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Bifurcations and catastrophes of a two-degrees-of-freedom nonlinear model simulating the buckling and postbuckling of rectangular plates
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Journal of the Franklin Institute 344 (2007) 463–488 www.elsevier.com/locate/jfranklin
Bifurcations and catastrophes of a two-degrees-of-freedom nonlinear model simulating the buckling and postbuckling of rectangular plates Dimitris S. Sophianopoulos Department of Civil Engineering, University of Thessaly, Pedion Areos, 38 334 Volos, Greece Received 9 February 2006; accepted 9 February 2006
Abstract The nonlinear buckling and postbuckling behavior of rectangular plates in symmetric and antisymmetric modes is re-examined, in the context of Bifurcation and Catastrophe Theories, using a two-degrees-of-freedom model, which has been adopted for the same purpose in the pioneer literature. At first the perfect system is dealt with in detail, symbolically utilizing the exact as well as the approximate equilibrium equations, the latter being products of a universal unfolding of the original total potential energy function. Conditions for the existence of remote secondary bifurcations are fully assessed and the stability of critical states is determined, revealing sudden qualitative changes in the postbuckling response of the perfect system, which have been also reported for the actual continuous structural system—the rectangular plate—using the von Ka´rman equations. Thereafter, the imperfection sensitivity is dealt with, introducing symmetric as well as asymmetric imperfections, considered as individual or consecutive perturbations of the perfect system. It is found that symmetry breaking bifurcations give birth to complicated cusp singularities, which may lead to unexpected jumps from one to two-mode remote postbuckling behavior. Finally, considering the general case of random imperfections, higher order two-mode singularities are revealed, mainly of the double-cusp catastrophe type, which have been also discovered in the
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E-mail address: [email protected]. 0016-0032/$30.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2006.02.012
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postbuckling response of rectangular plates, a fact validating the choice of the foregoing nonlinear simulation. r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Buckling and postbuckling; Rectangular plates; Modeling; Bifurcations; Catastrophes; Singularities; Cusps
1. Introduction One of the main advantages in the application of the methods of bifurcation theory to nonlinear stability problems is that in general it is possible to strongly reduce the dimension of the original (continuous) system to a low dimensional bifurcation system, mainly in the neighborhood of the critical points. The existence and possibility of such a reduction process follows from experimental evidence and has also been intuitively well-known to engineers for a long time. If, for example, an equilibrium position of an infinite dimensional system turns unstable, or for a small variation of the loading the system exhibits a totally different behavior, often only few (mostly one or two) modes generally determine the qualitative response of the structure dealt with. Evidently, the representation of the nonlinear bifurcation patterns of a continuous system via a simple model of a few degrees of freedom (DOFs) requires that the choice of the model is done in such a way, that the salient features of the real structure are well represented and properly captured. Moreover, even these models may under certain combinations of the foregoing parameters be associated with complicated bifurcation phenomena and a variety of distinct critical and postcritical responses, the study of which—through the simplified simulations—leads to the understanding of the behavior of the corresponding real system, without particular difficulties. This is the reason why numerous simple mechanical models have been introduced and thoroughly dealt with in the literature, in order to investigate all the indeed amazing multiplicity of static instabilities and bifurcations of structures, summarized in a very comprehensive manner in the pioneer work of Gioncu and Ivan [1]. More specifically, for structures with a distinct postcritical behavior (columns and plates under axial compression), there can be cases associated with remote secondary bifurcations arising in both perfect and imperfect configurations, associated with repetitive changes of the instability forms. In particular the buckling response of orthotropic rectangular plates was initially simulated via a 2-DoFs nonlinear system, developed by Stein (1959) and examined later by Supple and Chilver [2]. The fully nonlinear exact as well as approximate static bifurcation analysis of this particular model, either perfect or imperfect, is the main goal of the present paper, introducing also elements of the Theory of Catastrophes [3–5]. It is found that a variety of secondary bifurcations start from the main postcritical equilibrium configuration and that all related critical branching points in the case of no initial imperfections, under certain combinations of the foregoing parameters, are interconnected, belonging to a remote unstable path in the state-space. The introduction of a single either symmetric or antisymmetric imperfection (perturbation) leads to partial symmetry breaking, associated with independent as well as interconnected singularities of the dual-cusp form, which are all parametrically investigated and presented in detail. Finally, the fully imperfect system is found to exhibit instabilities only of the limit point
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type and is related to Catastrophe Manifolds of higher order two-mode singularities, resembling the double-cusp form, a finding also reported in the investigation of the buckling and postbuckling of rectangular plates and validating the chosen simulation. 2. Mathematical formulation, numerical results and discussion 2.1. Bifurcation theory approach We consider the generally initially imperfect 2-DOF system shown in Fig. 1(a, b), which consists of three weightless rigid bars of equal length l, pinned together at points 1 and 2, where two linear torsional springs of stiffness c are connected. Furthermore, these points are also connected with two freely sliding at their end support nonlinear extensional springs of stiffness k, while the midpoint of bar 12 is connected with a similarly supported nonlinear spring of stiffness 2k, all possessing a cubic helicoidal nonlinearity. This implies that the force Fi developing on each spring i ð¼ 1; 2; 3Þ due to a deformation di is equal to F i ¼ ki d3i . Considering as principal (generalized) coordinates (describing the DoFs of the system) the vertical deformations W1 and W2 and after introducing the following dimensionless parameters: P c ; b¼ 4 3 k‘ k‘ the nondimensionalized strain energy of the various spring elements can be written as 1 ðw1 e1 Þ þ ðw2 e2 Þ4 ðw1 e1 Þ4 þ 2 Uk ¼ þ ðw2 e2 Þ4 , 2 2 w1 ¼ W 1 =‘;
w2 ¼ W 2 =‘;
e1 ¼ W 1i =‘;
e2 ¼ W 2i =‘;
l¼
(1)
(2)
Fig. 1. Initial (a) and deformed (b) configuration of the 2-D.O.F. nonlinear model used in this study for the simulation of the buckling and postbuckling of (orthotropic) rectangular plates.
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1 U c ¼ bðW21 þ W22 Þ 2 where W1 ¼ ðarcsinðw1 Þ arcsinðe1 ÞÞ ðarcsinðw2 w1 Þ arcsinðe2 e1 ÞÞ
(3a) )
W2 ¼ ðarcsinðw2 Þ arcsinðe2 ÞÞ þ ðarcsinðw2 w1 Þ arcsinðe2 e1 ÞÞ while the dimensionless work of the external conservative force has the form: qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi O¼ l 1 e21 þ 1 ðe2 e1 Þ2 þ 1 e22 1 w21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðw2 w1 Þ2 1 w22 ,
(3b)
ð4Þ
where the quantity in the parenthesis represents the dimensionless horizontal deformation uA. Hence, the exact expression of the total potential of the system dealt with, is given by V T;exact ¼ U k þ U c þ O 2 3 ðe1 w1 Þ4 þ ðe2 w2 Þ4 þ 18 ðe1 þ e2 w1 w2 Þ4 6 7 0 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 6 7 2 2þ 2 6 7 1 e 1 ðe e Þ þ 1 e 1 2 1 2 6 7 B C 6 2B qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cl 7 6 7 @ A 1 2 7. 2 2 ¼ 6 1 w1 1 ðw1 w2 Þ 1 w2 6 7 26 7 6 7 6 2 !7 ðarcsin½e1 þ arcsin½e1 e2 arcsin½w1 arcsin½w1 w2 Þ 6 7 4 þb 5 2 þðarcsin½e1 e2 arcsin½e2 arcsin½w1 w2 þ arcsin½w2 Þ ð5Þ At this point we use the transformation of coordinates into symmetric and antisymmetric deformations, by setting w1 þ w2 w1 w2 ; wa ¼ ) w1 ¼ ws þ wa ; w2 ¼ ws wa ws ¼ (6a) 2 2 and thus it is also valid that e1 þ e2 e1 e2 es ¼ ; ea ¼ ) e1 ¼ es þ ea ; e2 ¼ es ea . (6b) 2 2 Substituting the above expressions into Eq. (5) the transformed exact potential becomes equal to: 3 2 2ðes ws Þ4 þ ðea þ es wa ws Þ4 þ ðea es wa þ ws Þ4 q q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 7 2 2 7 6 6 B 1 4e2a þ 1 ðea es Þ þ 1 ðea þ es Þ 1 4w2a C 7 6 2@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Al 7 7 16 transf 2 2 7. 6 V T;exact ¼ 6 1 ðwa ws Þ 1 ðwa þ ws Þ 7 26 !7 7 6 2 7 6 ðarcsin½2ea þ arcsin½ea es arcsin½2wa arcsin½wa ws Þ 5 4 þb 2 þðarcsin½ea þ es þ arcsin½2ea arcsin½wa þ ws arcsin½2wa Þ (7)
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Performing a Taylor expansion up to the 4th order of the transformed potential, we reach to the following approximate expression, which after cumbersome symbolic computations in Mathematica [6] is found to be a universal unfolding of the exact potential given in Eq. (5), according to the definitions of Catastrophe Theory [3–5]. 3 2 2ðes ws Þ4 þ ðea þ es wa ws Þ4 þ ðea es wa þ ws Þ4 ! 7 6 18e2a þ 2e2s 6ea wa ð6 þ 3w2a þ w2s Þ 23 es ws ð6 þ 3w2a þ w2s Þ 7 6 1 7 6þ b V transf ¼ 7. (8) 6 2 1 2 2 2 2 T;appr þ 18 ð6 þ 3wa þ ws Þ ð9wa þ ws Þ 7 26 5 4 1 2 2 2 2 2 2 þ 2 ð12ea þ 4es ð3wa þ ws Þð4 þ wa þ ws ÞÞl
2.1.1. Perfect system—approximate analysis Setting es ¼ ea ¼ 0, the equilibrium equations of the perfect system are determined according to the principles of nonlinear stability theory, given by: qV transf;p T;appr qws
qV transf;p T;appr qwa
¼
1 ws ½21w4a b þ 2w2a ð36 þ 24b þ 5w2s b 9lÞ 6 þ w2s ð48 þ ð8 þ w2s Þb 6l þ 12ðb lÞÞ ¼ 0
27w2a b þ w3a ð4 þ ð36 þ 7w2s Þb 9lÞ ¼ 2 5w4s b 2 þ ws ð12 þ 8b 3lÞ 6l þwa 18b þ ¼0 6
ð9aÞ
ð9bÞ
Except the trivial fundamental path ws ¼ wa ¼ 0, which is always a valid equilibrium configuration of the perfect system 8l40, it always exhibits two secondary equilibrium paths, characterized by symmetric and antisymmetric deformations and defined by analytical functions as follows: First secondary path, only symmetric deformation 1 2 48 þ b ½ws a0; wa ¼ 0 : l1 ¼ b þ ws (10) 6 2 þ w2s interesting the trivial path at the stable symmetric branching point (first critical point) C1, at which lappr cr1 ¼ l1 jws ¼0 ¼ b. The trivial path is up to this point stable, becoming thereafter (for higher load values) unstable. Second secondary path, only antisymmetric deformation 4 3b 2 ½ws ¼ 0; wa a0 : l2 ¼ 3b þ wa þ (11) 6 þ 9w2a 2 which intersects the trivial path at the unstable symmetric bifurcation point (second critical point) C2, where lappr cr2 ¼ l2 jwa ¼0 ¼ 3b, the presence of which does not affect the stability of the fundamental configuration. There also exists a third secondary path, associated with no symmetry, which can be evaluated only by the numerical solution of the strongly nonlinear equilibrium equations, since no analytical solution exists. In order that this third path intersects both the first and
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the second secondary paths, within the physical restrictions of the system i.e. for 1pwsp1, 0.5pwap0.5, after cumbersome symbolic computations, it is found that the inequality bp0.526748971 must hold. Selecting b ¼ 0:10 we thereafter determine the aforementioned third secondary path, being totally unstable, belonging to the configuration space and possessing two symmetry planes. This particular path intersects the first secondary path at the unstable symmetric branching points C 3 ; C 03 : ws ¼ 0:321682; wa ¼ 0; l ¼ 0:495279 and the second secondary path at the also unstable symmetric bifurcation points C 4 ; C 04 : ws ¼ 0; wa ¼ 0:197423; l ¼ 0:330395. These critical points affect the stability of the first and second secondary paths, since the former is stable between C3,C0 3 and thereafter becomes unstable, contrary to the latter, which is between C 4 ; C 04 unstable, becoming stable thereafter. All these equilibrium configurations along with the stability of the various branches are depicted in the 3D graphic representation given in Fig. 2. These findings reveal the existence of secondary bifurcations in remote postcritical domains of the system, associated with danger due to sudden changes—jumps of the instability form, which are in perfect agreement with existing results of the relevant literature [7,8], based on the buckling analysis of rectangular plates on the basis of the von Ka´rman equations.
Fig. 2. Equilibrium paths and critical points of the perfect system for b ¼ 0:10—approximate analysis.
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2.1.2. Perfect system—approximate analysis If instead of the approximate total potential energy function given in Eq. (8) one utilizes its corresponding exact value given in Eq. (7), for the nonlinear static stability analysis of the system, quite similar results are obtained. More specifically, except the fundamental configuration, the always existing first and second secondary paths (symmetric and antisymmetric, respectively) are given by: 1st secondary path, only symmetric deformation qffiffiffiffiffiffiffiffiffiffiffiffiffiffi b arcsin½ws l1 ¼ 4w2s 1 w2s þ , ws having a discontinuity at ws ¼ 0, but evidentlylexact cr1 ¼ limðl1 Þws !0 ¼ b, implying bifurcation point C1 and in the same manner 2nd secondary path, only antisymmetric deformation, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l2 ¼ w2a 2 1 4w2a ð1 þ w2a Þ þ 1 w2a ð1 þ 4w2a Þ 3 bðarcsin½wa þ arcsin½2wa Þ þ wa with a discontinuity at wa ¼ 0, but lexact cr2 ¼ limðl2 Þwa !0 ¼ 3b, implying branching point C2.
Fig. 3. Equilibrium paths and critical points of the perfect system for b ¼ 0:10—exact analysis.
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In order that the third secondary path (the one with no symmetries) intersects both the aforementioned first and second secondary paths the inequality bo4:69088 must be valid. Again, if b is set equal to 0.10, as in the previous section, this 3rd path intersects the 1st secondary one at the unstable symmetric branching points C 3 ; C 03 , for which ws ¼ 0:321205; wa ¼ 0; l ¼ 0:492627, and the 2nd secondary one at the also the unstable symmetric branching points C 4 ; C 04 having ws ¼ 0, wa ¼ 0:480114; l ¼ 0:428062. One may directly observe the minimal discrepancy of the results obtained between approximate and exact analyses, and confirm the same main instability changes, shown in the 3D equilibrium paths of Fig. 3. The existence of limit points on the 1st and 2nd secondary
Fig. 4. Fundamental and complementary equilibrium paths and corresponding critical points for the partially imperfect system of Case Ia: (a) for various values and (b) for two characteristic values of the initial symmetric imperfection es (solid lines ¼ stability, dashed lines ¼ instability).
Fig. 5. 3D representation of the critical points of the partially imperfect system of Case Ia ðb ¼ 0:10Þ.
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paths does not affect the overall response of the system, since they appear at large values of the deformations. Hence, the approximate analysis will be hereafter adopted for the nonlinear stability study of the corresponding imperfect system. 2.1.3. Imperfect system The set of nonlinear (approximate) equilibrium equations governing the response of the imperfect system are as follows: 9 8 8ðes ws Þ3 4ðea þ es wa ws Þ3 þ 4ðea es wa þ ws Þ3 > > = < 1 b3 ½6es ð2 þ w2a þ w2s Þ ws ð36ea wa þ ð6 þ 3w2a þ w2s Þð2 þ 7w2a þ w2w ÞÞ ¼ 0 > 2> ; : 2w ð2 þ 3w2 þ w2 Þl s a s (12a) 8 9 4ðea þ es wa ws Þ3 4ðea es wa þ ws Þ3 > > < =
1 e þ 6ea ð6 þ 9w2a þ w2s Þ þ wa 36 þ 27w4a 4es ws þ 16w2s þ 53 w4s þ 2w2a ð36 þ 7w2s Þ b ¼ 0. > 2 > : 6w ð2 þ 3w2 þ w2 Þl ; a
a
s
(12b) Two particular imperfection cases are dealt with below. The first is associated with partial (either only symmetric, Case (Ia) or only antisymmetric, Case (Ib) imperfections and the second with random values of es and ea different then zero (Case II). The value of b ¼ 0:10 is employed herein for all consequent analysis cases. Case Ia: es a0; ea ¼ 0. Eqs. (12a,b) admit two types of solutions. The one is related to zero antisymmetric deformation and leads to an analytical expression of the form: 1 48ðws es Þ3 3 ¼ þ b½w þ 6w 6e ; w ¼ 0 (13) limp s s a s 1 6ws 2 þ w2s
Fig. 6. Diagram of perturbed bifurcations (sensitivity to initial conditions) of the partially imperfect system of Case Ia ðb ¼ 0:10Þ.
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while the other, which may be evaluated only numerically, is associated with non zero values of both generalized coordinates. Eq. (13) substantially describes the primary as well as the complementary (physically not accepted) equilibrium configurations of the partially imperfect system with only symmetric imperfections, which are in fact perturbations of the fundamental as well as the 1st secondary path of the perfect system. Evidently, the complementary path, standing for values of es, ws with different signs, possesses a limit point (local extremum) S1, while the fundamental path, for which es, ws have the same signs is monotonically rising.
30
-
(a)
(b)
(c) Fig. 7. Projection of the equilibrium paths and critical points of the partially imperfect system of Case Ia, for es ¼ 0:025 and b ¼ 0:10.
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In order to investigate the perturbations of the bifurcations a thorough parametric study is carried out, by evaluating the critical points (and their stability) for a large range of values of es ð0:001pes p 0:50Þ. A partial symmetry breaking is revealed described in detail below. More specifically, the stable symmetric branching point C1 of the perfect system evolves to limit point S1 (as stated previously) and the system exhibits an independent dual-cusp catastrophe regardless of the value of es. Moreover, beyond this phenomenon, the unstable symmetric branching point C3 (on the 1st secondary path of the perfect system) gives birth to the also unstable symmetric bifurcation point C imp on the 3 monotonically rising fundamental path of the imperfect system. Its effect on the stability of the fundamental path is similar of the one of C3, since from the initial equilibrium configuration up to C imp the fundamental path is stable, becoming thereafter unstable. 3 Thus, a new dual-cusp singularity is revealed, initiated at C3, which is not independent, but connected to other dual-cusps, resulting from the perturbation of branching points C 03 and C2. Actually, for small values of jes j the complementary path exhibits (except S1) two unstable symmetric branching points, namely C 0imp and C imp 3 2 , which result from the perturbation of the also unstable symmetric bifurcation points C 03 and C2 of the 1st secondary and trivial path of the perfect system, respectively. As es increases absolutely, critical points C 0imp and C imp approach each other, and at jes j ¼ jecr s j ¼ 0:067 coincide and 3 2 then vanish. Thus, two more branches of interconnected perturbed bifurcations appear,
Fig. 8. 3D equilibrium paths and critical points of the partially imperfect system of Case Ia, for es ¼ 0:025 and b ¼ 0:10.
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starting from C2 and ending at C3, C 03 , to connect there with the dual-cusp due to the perturbation of C3. Consequently three interconnected dual cusps are present, a finding firstly reported in the literature [9,10]. As far as stability is concerned, only point C 0imp 3 changes, in conjunction with S1, the stability of the complementary path, while the presence of C imp does not affect the system’s overall stability. Of course, the presence of 2 imp critical points C imp and C 0imp is due to the intersection of the equilibrium branches on 2 , C3 3 which these exist with other secondary paths, the latter being the symmetry breaking perturbation of the third secondary path of the perfect system. In fact, the system besides the fundamental and complementary paths (on the [ws,l] configuration plane) possesses also two secondary branches on the configuration space [ws, wa, l], being curves having the
Fig. 9. As in Fig. 4, but for Case Ib.
Fig. 10. As in Fig. 5, but for Case Ib.
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above plane as a plane of symmetry. From these paths, the one is an open curve and exhibits two limit points S1 and S01 , being perturbations of branching points C4 and C 04 of the perfect system, respectively, creating two additional dual cusps on the configuration space. Their presence changes the stability of this particular path, which is unstable between them and unstable beyond them. Contrary, the 2nd secondary path, although possessing the same symmetry plane as the 1st one, is a closed curve connecting critical points C imp and C 0imp 2 3 , and is unstable throughout its length. All the preceding general findings are presented graphically throughout Figs. 4–8. More analytically, in Fig. 4a one may perceive the fundamental as well as the complementary equilibrium paths with their critical points for various values of the symmetric initial imperfection, while in Fig. 4b the stability of two characteristic such paths is presented (with the dashed line implying instability). In Fig. 5 all the critical points of the foregoing parametric study are depicted in 3D form, while the diagram of perturbed bifurcations on the [es,l] plane is shown in Fig. 6. As a paradigm, the case of a system with es ¼ 0:025 is dealt with in detail, and its corresponding equilibrium paths with all the critical points and stability branches are shown in 2D and 3D in Figs. 7(a,b,c) and 8, respectively. Case Ib: es ¼ 0; ea a0: For this sub-case also, Eqs. (12a,b) admit two type of solutions. The one related with zero symmetric deformation is given by the following analytical expression: 1 8ðwa ea Þ3 imp 3 ws ¼ 0; l2 ¼ þ 9b½wa þ 2wa 2ea (14) 6wa 2 þ 3w2a while the other, that may be evaluated only numerically, is associated with non zero values of both generalized coordinates, as in the previous sub-case. Eq. (14) stands for the fundamental as well as the complementary (physically not accepted) equilibrium configurations of the partially imperfect system with only antisymmetric imperfections. The fundamental paths correspond to ea, wa with equal signs, while the complementary ones, which always exhibit a local minimum—limit point S2, to ea, wa with opposite signs. Both these paths are the perturbations of the trivial as well as the 2nd secondary paths of the perfect system.
Fig. 11. As in Fig. 6, but for Case Ib.
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Fig. 12. As in Fig. 7, but for Case Ib with ea ¼ 0:025 and b ¼ 0:10.
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As already performed for Case Ia, a thorough parametric study is carried out, in order to investigate the perturbations of the bifurcations, by evaluating the critical points (and their stability) for a large range of values of ea ð0:001pea p 0:345Þ. The introduction of a single perturbation on the perfect system, namely of the antisymmetric initial imperfection, leads also to a partial symmetry breaking. More analytically, the unstable symmetric branching point C2 evolves into limit point S2 and thus the system exhibits an independent dual-cusp form singularity, regardless of the value of ea. The complementary equilibrium path, to which S2 belongs possesses also another critical point, the unstable symmetric branching point C 0imp 4 , being the perturbation of the also the unstable symmetric branching C 04 of the 2nd secondary path of the perfect system. The new dual cusp initiated from C 04 is not independent, but as it will be shown below, is connected with two more singularities of the same form, which are originated from critical points C1 and C4. Evidently, for small values of jea j the monotonically rising (and initially stable) fundamental equilibrium path of the imperfect system possesses two critical branching points, the stable symmetric bifurcation point C imp 1 , product of the perturbation of critical point C1 (of the corresponding perfect system), as well as the unstable bifurcation point C imp 4 , emerging from the perturbation of critical point C4 (of the 2nd secondary path of the perfect system). As ea increases absolutely, points C imp and C imp approach each other and 1 4
Fig. 13. As in Fig. 8, but for Case Ib with ea ¼ 0:025 and b ¼ 0:10.
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at jecr a j ¼ 0:101163 coincide and vanish, and hence the fundamental path does not have any critical point from this of value of jea j on. The two dual cusps resulting from C1 and C4 are consequently interconnected, and reach points C4, C 04 , where they meet the dual cusp due to C 04 . Hence, three dual cusp singularities are connected to each other, a very important new finding fist reported in the literature [9,10], similar to the one obtained for Case Ia. The presence of critical point C 0imp on the complementary path is the one separating it 4 between a stable and an unstable branch, while the initial position of limit point S2 on this specific path is not affecting its stability. Contrary, the branching points C imp and C imp 1 4 become (if existing) the limits between which the fundamental part is unstable and stable beyond them. According to the theorems of nonlinear stability, as also for Case Ia, the existence of bifurcation points on the primary and complementary paths, on the [wa,l] plane, is ought to the presence of two secondary paths in the configuration space, which have the above plane as a symmetry one, and are in fact symmetry braking perturbations of the 3rd secondary path of the perfect system. The 1st of these secondary paths passes across point C 0imp 4 , is an open curve, contains only unstable equilibria and has no other critical points. On the other hand, the 2nd secondary path is a closed curve connecting points C imp and 1
Fig. 14. Projections (a) and 3D representations (b) of the equilibrium paths of the fully imperfect system with es ¼ ea ¼ 0:025 (grey lines ¼ stability, black lines ¼ instability).
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Fig. 15. Example of the evolution of the critical points of the system due to imperfections ðb ¼ 0:10Þ.
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C imp and possesses also two limit point S2 and S02 , which are perturbations of points C3 4 and C 03 , respectively, creating thus two more independent dual cusps in the configuration space. These cusps start from the above points and asymptotically tend to that state of the imp system, for which at ea ¼ ecr and C imp coincide and then a the critical points C 1 4 disappear. The presence of the aforementioned limit point on this particular path affects its stability, since it is stable from C imp until these, and thereafter, until C imp 1 4 , unstable. All the above findings concerning the foregoing case are presented in graphical form throughout Figs. 9–13. Particularly, in Fig. 9a one may perceive the fundamental as well as the complementary equilibrium paths with their critical points for various values of the antisymmetric initial imperfection, while in Fig. 9b the stability of two characteristic such paths is presented (with the dashed line implying instability). Moreover, in Fig. 10 all the critical points of the foregoing parametric study are depicted in 3D form, while the
Fig. 16. Catastrophe Manifold Surface of the partially imperfect system of Case Ia: (a) of the first kind, (b) of the second kind.
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Fig. 17. Characteristic 3d views of the Catastrophe Manifolds of Fig. 16: (a) of the 1st kind, (b) of the 2nd kind.
diagram of perturbed bifurcations on the [ea,l] plane is shown in Fig. 11. Adopting lastly the value of ea ¼ 0:025 as an example, the equilibrium paths and their correspond stability branches along with all critical points are determined and shown in Figs. 12(a,b,c,) and 13 in 2D and 3D form respectively. Case II: Random imperfections: es ; ea a0. This last case of random imperfections, being in substance a double perturbation of the perfect system, cannot be systematically dealt with, within the Context of Nonlinear Stability and Bifurcation Theory, due to the strongly nonlinear nature of the governing equilibrium equations, but as it will be shown below, it may be comprehensively investigated using principles of the Theory of Catastrophes. Hence, the absence of analytical solutions for the various equilibrium configurations leads to their individual numerical evaluation (including critical points and stability) for each combinations of es, ea separately. For reason of comparison with the findings related to the perfect and partially imperfect system given earlier, the case of es ¼ ea ¼ 0:025 is studied and the corresponding equilibria, critical states as well as their overall stability is determined and presented graphically in Fig. 14. The system under this combination of parameters has been found to exhibit four (4) independent paths in the configuration space, possessing in total six (6) limit points, which are the evolutions—perturbations of the corresponding critical points of the partially imperfect system of either Cases Ia and Ib. These evolutions may be comprehensively perceived in the schematic diagram of Fig. 15, starting from the branching points of the perfect system. 2.2. Catastrophe theory approach 2.2.1. General considerations The partially imperfect system (Case Ia, Ib) is characterized by two main (active) coordinates, namely ws and wa, and also two control parameters, i.e. the loading l and the corresponding initial imperfection (either es or ea). In the sequel, its total potential does not
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belong to one of the seven elementary Catastrophes firstly introduced by Arnold [11]. Furthermore, the fully imperfect system, has two active coordinates and three control parameters but cannot be classified to one of the above 7 elementary singularities, since its potential differs tremendously from those of the hyperbolic or elliptic umbilic (which belong to these elementary catastrophes and have the same number of active variables and control parameters). Hence, no obvious classification or correlation exists and one must resort to the reconstruction of the Catastrophe Manifolds and Bifurcations Sets of the system, either partially or fully initially imperfect, and to the qualitative/quantitative study (to the most possible extent) of their salient features and properties. 2.2.2. Catastrophe manifolds (M) and bifurcations sets (Bs) According to the definitions and principles of the Theory of Catastrophes, the Catastrophe Manifolds, being the loci of equilibria, are in general hypersurfaces resulting from the equilibrium equations, while the Bifurcations Sets are generally also
Fig. 18. As in Fig. 16, but for Case Ib.
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hypersurfaces that require also the zeroing of the main stability determinant (Hessian) of the system. More specifically, for the partially imperfect system, employing advanced symbolic as well as graphical functions embedded in modern commercial mathematical software (Mathematica [6]) one may reconstruct two specific kinds of Catastrophe Manifold Surfaces, resulting from the elimination of one control parameter from Eqs. (12a,b), either the initial imperfection (es or ea) or the loading l. The resulting surfaces except their profound partial symmetry features, all exhibit the five basic characteristics of the dual-cusp Catastrophe, i.e. bi-modality, inaccessibility, sudden jumps, divergence and hysterisis. These properties are clearly depicted in Figs. 16–19, where various 3D views of the Catastrophe Manifolds of both aforementioned kinds is depicted. Moreover, in order to validate the results of the preceding Bifurcation Theory approach, the Catastrophe Manifold Surfaces of the 2nd kind (in the [ws, wa, imperfection] space) are redrawn, for both Cases Ia and Ib, the critical planes jes j and jea j are depicted in Fig. 20, while also the planes es ¼ 0:025 and ea ¼ 0:025 with all the critical points evaluated in the parametric studies are shown. From this particular figure the findings of the Bifurcation analysis are fully assessed, while the existence of independent and interconnected dual cusp singularities is comprehensively presented. In the same manner, for the partially imperfect system, the Bifurcation Sets are reconstructed as surfaces in the [ws, wa, imperfection] plane and presented in Fig. 21, revealing an extremely complicated pattern but nevertheless containing also the main characteristics of dual-cusps. This is not the case however for the Catastrophe Manifold of the fully imperfect system, since the matter concerns hypersufaces in a four dimensional space of the form Mðx; y; ws ; wa ; es ; ea Þ, and only projections of these on some mathematically convenient 3D space may be visualized. In doing these, for arbitrary values of es or ea , eliminating from the equilibrium equations the control parameter l, one may rather easily acquire
Fig. 19. As in Fig. 17, but for Case Ib.
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projections of the aforementioned hypersurfaces on either [ws,wa,ea] or [ws,wa,ws] spaces. The surfaces produced, as characteristically shown in Fig. 22, contain the main feature of full symmetry breaking and are related to multiple cusps, folds and limit points, being quite similar with the double-cusp singularity type (X9 as in [11]), which has been reported to be associated with several problems dealing with the buckling and postbuckling of rectangular plates [12,13], a finding validating the chosen simulation. Finally, the results of the Bifurcation Theory approach are fully validated, by reproducing sections of the Manifold projections for es ¼ 0:025 and ea ¼ 0:025 with the ea ¼ 0:025 and es ¼ 0:025 planes, respectively, as shown in Fig. 23.
Fig. 20. Catastrophe Manifold of the 2nd kind of the partially imperfect system. Left hand side (Case Ia): (a) cr surface with critical points, (b) surface with critical planes ecr s and es , (c) section with the es ¼ 0:025 plane (identical with Fig. 7a, Right-hand side (Case Ib): (d) surface with critical points, (e) surface with critical planes ecr a and ecr a , (c) section with the ea ¼ 0:025 plane (identical with Fig. 12a).
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3. Conclusions The most important conclusions drawn from this study are the following: The nonlinear 2-D.O.F. system dealt with in detail, exhibits an extraordinary postbuckling behavior, which for certain combinations of the parameters involved is related to remote postbuckling bifurcations, which connect/couple the secondary symmetric and antisymmetric equilibrium configurations and strongly affect their stability. Therefore, the danger of sudden, not expected transition (jump) from one to two mode buckling and visa versa is significant, for both perfect and partially initially imperfect system. This finding is of great importance for structural design purposes, while the possibility of the occurrence of such phenomena has been reported in the literature, for the buckling and postbuckling of rectangular plates, a fact validating the correct choice of the proposed model and the accuracy of its exact as well as approximate bifurcational analysis. The introduction of initial imperfections (partial perturbations) in the system (either only symmetric or only antisymmetric) leads to partial symmetry breaking, which
Fig. 21. Bifurcational Sets of the partially imperfect system for Case Ia (a) and Case Ib (b).
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Fig. 22. Various characteristic projections of the Catastrophe Manifold Hypersurface of the fully imperfect system ðb ¼ 0:10Þ.
gives birth to complicated bifurcations, related to independent and interconnected dual-cusp singularities, and the danger of mode jumping (although not usually related to imperfect systems) is visible. The fully nonlinear static buckling analysis produced results in full accordance with the ones gained from the Catastrophe Theory approach, the latter indicating singularities that cannot be classified in one of the seven elementary ones, although two active coordinates and two control parameters are involved. Finally, introducing an arbitrary full initial imperfection in the system, considered either as an overall perturbation of the perfect one or as a single perturbation of the partially imperfect one, leads to the evolution of all critical points to limit point ones, combined with total symmetry breaking. Through a characteristic numerical application, the full accordance of the findings of both Bifurcational and Catastrophe Theory approaches was demonstrated, the latter revealing the existence of a complicated higher order two-mode singularity, containing the main features of the double-cusp, which has been reported being related with the buckling of plates. Hence, once again the accuracy and validity of the proposed simulation is confirmed.
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Fig. 23. Projections of the Catastrophe Manifold Hypersurface of the fully imperfect system. Left-hand side (a,b,c): On the [ws,wa,ea] space for es ¼ 0:025 and section with the ea ¼ 0:025 plane, Right-hand side (d,e,f): On the [ws,wa,es] space for ea ¼ 0:025 and section with the es ¼ 0:025 plane—full accordance with Fig. 14c,d.
References [1] V. Gioncu, M. Ivan, Theory of critical and postcritical behavior of elastic structures, Editura Academiei Republicii Socialiste Romania, 1984. [2] W.J. Supple, A.H. Chilver, Elastic post-buckling of compressed rectangular plates, in: A.H. Chilver (Ed.), Thin-Walled Structures, Chatto & Windus, London, 1967. [3] H. Troger, A. Steindl, Nonlinear Stability and Bifurcation Theory: An Introduction for Engineers and Scientists, Springer, Wien/NY, 1991. [4] R. Gilmore, Catastrophe Theory for Scientists and Engineers, Dover, UK, 1993. [5] T. Poston, I. Stewart, Catastrophe Theory and its Applications, Pitman, London, 1978. [6] T.B. Bahder, Mathematica for Scientists and Engineers, Addison-Wesley, New York, 1995. [7] T. Nakamura, K. Uetani, The secondary buckling and post-secondary buckling behaviors of rectangular plates, Int. J. Mech. Sci. 21 (1979) 256–286. [8] P.R. Everall, G.W. Hunt, Anrold tongue predictions of secondary buckling in thin elastic plates, J. Mech. Phys. Solids 47 (1999) 2187–2206.
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[9] D.S. Sophianopoulos, L. Argyropoulou, E. Evagorou, V. Marinithou, Secondary bifurcations in postrcritical domains of a two-degree-of-freedom nonlinear system with elements of catastrophe theory, in: A.N. Kounadis, C. Povidakis, G. Exadaktylos (Eds.), Proceedings of the Seventh National Congress on Mechanics, Chania, Crete, Greece, June 24–26, 2004, Technical University of Crete, vol. I, pp. 244—249. [10] D.S. Sophianopoulos, Bifurcations and Catastrophes of two-degree-of-freedom nonlinear model simulating the buckling and postbuckling of rectangular plates, in: Proceedings of the First International Conference on Modeling, Simulation and Applied Optimization, Sharjah, U.A.E., February 1–3, 2005 (Paper ID: 132, Proc. in CD-ROM, ICMSAO 2005, ISBN: 9948-427-00-9). [11] V.I. Arnold, Catastrophe Theory, third ed., Springer, Berlin, 1992. [12] J.M.T. Thompson, Instabilities and Catastrophes in Science and Engineering, Wiley, New York, 1982. [13] J.M.T. Thompson, G.W. Hunt, Elastic Instability Phenomena, Wiley, Chichester, 1984.
Journal of the Franklin Institute 344 (2007) 463–488 www.elsevier.com/locate/jfranklin
Bifurcations and catastrophes of a two-degrees-of-freedom nonlinear model simulating the buckling and postbuckling of rectangular plates Dimitris S. Sophianopoulos Department of Civil Engineering, University of Thessaly, Pedion Areos, 38 334 Volos, Greece Received 9 February 2006; accepted 9 February 2006
Abstract The nonlinear buckling and postbuckling behavior of rectangular plates in symmetric and antisymmetric modes is re-examined, in the context of Bifurcation and Catastrophe Theories, using a two-degrees-of-freedom model, which has been adopted for the same purpose in the pioneer literature. At first the perfect system is dealt with in detail, symbolically utilizing the exact as well as the approximate equilibrium equations, the latter being products of a universal unfolding of the original total potential energy function. Conditions for the existence of remote secondary bifurcations are fully assessed and the stability of critical states is determined, revealing sudden qualitative changes in the postbuckling response of the perfect system, which have been also reported for the actual continuous structural system—the rectangular plate—using the von Ka´rman equations. Thereafter, the imperfection sensitivity is dealt with, introducing symmetric as well as asymmetric imperfections, considered as individual or consecutive perturbations of the perfect system. It is found that symmetry breaking bifurcations give birth to complicated cusp singularities, which may lead to unexpected jumps from one to two-mode remote postbuckling behavior. Finally, considering the general case of random imperfections, higher order two-mode singularities are revealed, mainly of the double-cusp catastrophe type, which have been also discovered in the
Corresponding author. Tel.: +30 6937017658; fax: +30 2106521793.
E-mail address: [email protected]. 0016-0032/$30.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2006.02.012
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postbuckling response of rectangular plates, a fact validating the choice of the foregoing nonlinear simulation. r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Buckling and postbuckling; Rectangular plates; Modeling; Bifurcations; Catastrophes; Singularities; Cusps
1. Introduction One of the main advantages in the application of the methods of bifurcation theory to nonlinear stability problems is that in general it is possible to strongly reduce the dimension of the original (continuous) system to a low dimensional bifurcation system, mainly in the neighborhood of the critical points. The existence and possibility of such a reduction process follows from experimental evidence and has also been intuitively well-known to engineers for a long time. If, for example, an equilibrium position of an infinite dimensional system turns unstable, or for a small variation of the loading the system exhibits a totally different behavior, often only few (mostly one or two) modes generally determine the qualitative response of the structure dealt with. Evidently, the representation of the nonlinear bifurcation patterns of a continuous system via a simple model of a few degrees of freedom (DOFs) requires that the choice of the model is done in such a way, that the salient features of the real structure are well represented and properly captured. Moreover, even these models may under certain combinations of the foregoing parameters be associated with complicated bifurcation phenomena and a variety of distinct critical and postcritical responses, the study of which—through the simplified simulations—leads to the understanding of the behavior of the corresponding real system, without particular difficulties. This is the reason why numerous simple mechanical models have been introduced and thoroughly dealt with in the literature, in order to investigate all the indeed amazing multiplicity of static instabilities and bifurcations of structures, summarized in a very comprehensive manner in the pioneer work of Gioncu and Ivan [1]. More specifically, for structures with a distinct postcritical behavior (columns and plates under axial compression), there can be cases associated with remote secondary bifurcations arising in both perfect and imperfect configurations, associated with repetitive changes of the instability forms. In particular the buckling response of orthotropic rectangular plates was initially simulated via a 2-DoFs nonlinear system, developed by Stein (1959) and examined later by Supple and Chilver [2]. The fully nonlinear exact as well as approximate static bifurcation analysis of this particular model, either perfect or imperfect, is the main goal of the present paper, introducing also elements of the Theory of Catastrophes [3–5]. It is found that a variety of secondary bifurcations start from the main postcritical equilibrium configuration and that all related critical branching points in the case of no initial imperfections, under certain combinations of the foregoing parameters, are interconnected, belonging to a remote unstable path in the state-space. The introduction of a single either symmetric or antisymmetric imperfection (perturbation) leads to partial symmetry breaking, associated with independent as well as interconnected singularities of the dual-cusp form, which are all parametrically investigated and presented in detail. Finally, the fully imperfect system is found to exhibit instabilities only of the limit point
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type and is related to Catastrophe Manifolds of higher order two-mode singularities, resembling the double-cusp form, a finding also reported in the investigation of the buckling and postbuckling of rectangular plates and validating the chosen simulation. 2. Mathematical formulation, numerical results and discussion 2.1. Bifurcation theory approach We consider the generally initially imperfect 2-DOF system shown in Fig. 1(a, b), which consists of three weightless rigid bars of equal length l, pinned together at points 1 and 2, where two linear torsional springs of stiffness c are connected. Furthermore, these points are also connected with two freely sliding at their end support nonlinear extensional springs of stiffness k, while the midpoint of bar 12 is connected with a similarly supported nonlinear spring of stiffness 2k, all possessing a cubic helicoidal nonlinearity. This implies that the force Fi developing on each spring i ð¼ 1; 2; 3Þ due to a deformation di is equal to F i ¼ ki d3i . Considering as principal (generalized) coordinates (describing the DoFs of the system) the vertical deformations W1 and W2 and after introducing the following dimensionless parameters: P c ; b¼ 4 3 k‘ k‘ the nondimensionalized strain energy of the various spring elements can be written as 1 ðw1 e1 Þ þ ðw2 e2 Þ4 ðw1 e1 Þ4 þ 2 Uk ¼ þ ðw2 e2 Þ4 , 2 2 w1 ¼ W 1 =‘;
w2 ¼ W 2 =‘;
e1 ¼ W 1i =‘;
e2 ¼ W 2i =‘;
l¼
(1)
(2)
Fig. 1. Initial (a) and deformed (b) configuration of the 2-D.O.F. nonlinear model used in this study for the simulation of the buckling and postbuckling of (orthotropic) rectangular plates.
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1 U c ¼ bðW21 þ W22 Þ 2 where W1 ¼ ðarcsinðw1 Þ arcsinðe1 ÞÞ ðarcsinðw2 w1 Þ arcsinðe2 e1 ÞÞ
(3a) )
W2 ¼ ðarcsinðw2 Þ arcsinðe2 ÞÞ þ ðarcsinðw2 w1 Þ arcsinðe2 e1 ÞÞ while the dimensionless work of the external conservative force has the form: qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi O¼ l 1 e21 þ 1 ðe2 e1 Þ2 þ 1 e22 1 w21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðw2 w1 Þ2 1 w22 ,
(3b)
ð4Þ
where the quantity in the parenthesis represents the dimensionless horizontal deformation uA. Hence, the exact expression of the total potential of the system dealt with, is given by V T;exact ¼ U k þ U c þ O 2 3 ðe1 w1 Þ4 þ ðe2 w2 Þ4 þ 18 ðe1 þ e2 w1 w2 Þ4 6 7 0 qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 6 7 2 2þ 2 6 7 1 e 1 ðe e Þ þ 1 e 1 2 1 2 6 7 B C 6 2B qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cl 7 6 7 @ A 1 2 7. 2 2 ¼ 6 1 w1 1 ðw1 w2 Þ 1 w2 6 7 26 7 6 7 6 2 !7 ðarcsin½e1 þ arcsin½e1 e2 arcsin½w1 arcsin½w1 w2 Þ 6 7 4 þb 5 2 þðarcsin½e1 e2 arcsin½e2 arcsin½w1 w2 þ arcsin½w2 Þ ð5Þ At this point we use the transformation of coordinates into symmetric and antisymmetric deformations, by setting w1 þ w2 w1 w2 ; wa ¼ ) w1 ¼ ws þ wa ; w2 ¼ ws wa ws ¼ (6a) 2 2 and thus it is also valid that e1 þ e2 e1 e2 es ¼ ; ea ¼ ) e1 ¼ es þ ea ; e2 ¼ es ea . (6b) 2 2 Substituting the above expressions into Eq. (5) the transformed exact potential becomes equal to: 3 2 2ðes ws Þ4 þ ðea þ es wa ws Þ4 þ ðea es wa þ ws Þ4 q q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 7 2 2 7 6 6 B 1 4e2a þ 1 ðea es Þ þ 1 ðea þ es Þ 1 4w2a C 7 6 2@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Al 7 7 16 transf 2 2 7. 6 V T;exact ¼ 6 1 ðwa ws Þ 1 ðwa þ ws Þ 7 26 !7 7 6 2 7 6 ðarcsin½2ea þ arcsin½ea es arcsin½2wa arcsin½wa ws Þ 5 4 þb 2 þðarcsin½ea þ es þ arcsin½2ea arcsin½wa þ ws arcsin½2wa Þ (7)
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Performing a Taylor expansion up to the 4th order of the transformed potential, we reach to the following approximate expression, which after cumbersome symbolic computations in Mathematica [6] is found to be a universal unfolding of the exact potential given in Eq. (5), according to the definitions of Catastrophe Theory [3–5]. 3 2 2ðes ws Þ4 þ ðea þ es wa ws Þ4 þ ðea es wa þ ws Þ4 ! 7 6 18e2a þ 2e2s 6ea wa ð6 þ 3w2a þ w2s Þ 23 es ws ð6 þ 3w2a þ w2s Þ 7 6 1 7 6þ b V transf ¼ 7. (8) 6 2 1 2 2 2 2 T;appr þ 18 ð6 þ 3wa þ ws Þ ð9wa þ ws Þ 7 26 5 4 1 2 2 2 2 2 2 þ 2 ð12ea þ 4es ð3wa þ ws Þð4 þ wa þ ws ÞÞl
2.1.1. Perfect system—approximate analysis Setting es ¼ ea ¼ 0, the equilibrium equations of the perfect system are determined according to the principles of nonlinear stability theory, given by: qV transf;p T;appr qws
qV transf;p T;appr qwa
¼
1 ws ½21w4a b þ 2w2a ð36 þ 24b þ 5w2s b 9lÞ 6 þ w2s ð48 þ ð8 þ w2s Þb 6l þ 12ðb lÞÞ ¼ 0
27w2a b þ w3a ð4 þ ð36 þ 7w2s Þb 9lÞ ¼ 2 5w4s b 2 þ ws ð12 þ 8b 3lÞ 6l þwa 18b þ ¼0 6
ð9aÞ
ð9bÞ
Except the trivial fundamental path ws ¼ wa ¼ 0, which is always a valid equilibrium configuration of the perfect system 8l40, it always exhibits two secondary equilibrium paths, characterized by symmetric and antisymmetric deformations and defined by analytical functions as follows: First secondary path, only symmetric deformation 1 2 48 þ b ½ws a0; wa ¼ 0 : l1 ¼ b þ ws (10) 6 2 þ w2s interesting the trivial path at the stable symmetric branching point (first critical point) C1, at which lappr cr1 ¼ l1 jws ¼0 ¼ b. The trivial path is up to this point stable, becoming thereafter (for higher load values) unstable. Second secondary path, only antisymmetric deformation 4 3b 2 ½ws ¼ 0; wa a0 : l2 ¼ 3b þ wa þ (11) 6 þ 9w2a 2 which intersects the trivial path at the unstable symmetric bifurcation point (second critical point) C2, where lappr cr2 ¼ l2 jwa ¼0 ¼ 3b, the presence of which does not affect the stability of the fundamental configuration. There also exists a third secondary path, associated with no symmetry, which can be evaluated only by the numerical solution of the strongly nonlinear equilibrium equations, since no analytical solution exists. In order that this third path intersects both the first and
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the second secondary paths, within the physical restrictions of the system i.e. for 1pwsp1, 0.5pwap0.5, after cumbersome symbolic computations, it is found that the inequality bp0.526748971 must hold. Selecting b ¼ 0:10 we thereafter determine the aforementioned third secondary path, being totally unstable, belonging to the configuration space and possessing two symmetry planes. This particular path intersects the first secondary path at the unstable symmetric branching points C 3 ; C 03 : ws ¼ 0:321682; wa ¼ 0; l ¼ 0:495279 and the second secondary path at the also unstable symmetric bifurcation points C 4 ; C 04 : ws ¼ 0; wa ¼ 0:197423; l ¼ 0:330395. These critical points affect the stability of the first and second secondary paths, since the former is stable between C3,C0 3 and thereafter becomes unstable, contrary to the latter, which is between C 4 ; C 04 unstable, becoming stable thereafter. All these equilibrium configurations along with the stability of the various branches are depicted in the 3D graphic representation given in Fig. 2. These findings reveal the existence of secondary bifurcations in remote postcritical domains of the system, associated with danger due to sudden changes—jumps of the instability form, which are in perfect agreement with existing results of the relevant literature [7,8], based on the buckling analysis of rectangular plates on the basis of the von Ka´rman equations.
Fig. 2. Equilibrium paths and critical points of the perfect system for b ¼ 0:10—approximate analysis.
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2.1.2. Perfect system—approximate analysis If instead of the approximate total potential energy function given in Eq. (8) one utilizes its corresponding exact value given in Eq. (7), for the nonlinear static stability analysis of the system, quite similar results are obtained. More specifically, except the fundamental configuration, the always existing first and second secondary paths (symmetric and antisymmetric, respectively) are given by: 1st secondary path, only symmetric deformation qffiffiffiffiffiffiffiffiffiffiffiffiffiffi b arcsin½ws l1 ¼ 4w2s 1 w2s þ , ws having a discontinuity at ws ¼ 0, but evidentlylexact cr1 ¼ limðl1 Þws !0 ¼ b, implying bifurcation point C1 and in the same manner 2nd secondary path, only antisymmetric deformation, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l2 ¼ w2a 2 1 4w2a ð1 þ w2a Þ þ 1 w2a ð1 þ 4w2a Þ 3 bðarcsin½wa þ arcsin½2wa Þ þ wa with a discontinuity at wa ¼ 0, but lexact cr2 ¼ limðl2 Þwa !0 ¼ 3b, implying branching point C2.
Fig. 3. Equilibrium paths and critical points of the perfect system for b ¼ 0:10—exact analysis.
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In order that the third secondary path (the one with no symmetries) intersects both the aforementioned first and second secondary paths the inequality bo4:69088 must be valid. Again, if b is set equal to 0.10, as in the previous section, this 3rd path intersects the 1st secondary one at the unstable symmetric branching points C 3 ; C 03 , for which ws ¼ 0:321205; wa ¼ 0; l ¼ 0:492627, and the 2nd secondary one at the also the unstable symmetric branching points C 4 ; C 04 having ws ¼ 0, wa ¼ 0:480114; l ¼ 0:428062. One may directly observe the minimal discrepancy of the results obtained between approximate and exact analyses, and confirm the same main instability changes, shown in the 3D equilibrium paths of Fig. 3. The existence of limit points on the 1st and 2nd secondary
Fig. 4. Fundamental and complementary equilibrium paths and corresponding critical points for the partially imperfect system of Case Ia: (a) for various values and (b) for two characteristic values of the initial symmetric imperfection es (solid lines ¼ stability, dashed lines ¼ instability).
Fig. 5. 3D representation of the critical points of the partially imperfect system of Case Ia ðb ¼ 0:10Þ.
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paths does not affect the overall response of the system, since they appear at large values of the deformations. Hence, the approximate analysis will be hereafter adopted for the nonlinear stability study of the corresponding imperfect system. 2.1.3. Imperfect system The set of nonlinear (approximate) equilibrium equations governing the response of the imperfect system are as follows: 9 8 8ðes ws Þ3 4ðea þ es wa ws Þ3 þ 4ðea es wa þ ws Þ3 > > = < 1 b3 ½6es ð2 þ w2a þ w2s Þ ws ð36ea wa þ ð6 þ 3w2a þ w2s Þð2 þ 7w2a þ w2w ÞÞ ¼ 0 > 2> ; : 2w ð2 þ 3w2 þ w2 Þl s a s (12a) 8 9 4ðea þ es wa ws Þ3 4ðea es wa þ ws Þ3 > > < =
1 e þ 6ea ð6 þ 9w2a þ w2s Þ þ wa 36 þ 27w4a 4es ws þ 16w2s þ 53 w4s þ 2w2a ð36 þ 7w2s Þ b ¼ 0. > 2 > : 6w ð2 þ 3w2 þ w2 Þl ; a
a
s
(12b) Two particular imperfection cases are dealt with below. The first is associated with partial (either only symmetric, Case (Ia) or only antisymmetric, Case (Ib) imperfections and the second with random values of es and ea different then zero (Case II). The value of b ¼ 0:10 is employed herein for all consequent analysis cases. Case Ia: es a0; ea ¼ 0. Eqs. (12a,b) admit two types of solutions. The one is related to zero antisymmetric deformation and leads to an analytical expression of the form: 1 48ðws es Þ3 3 ¼ þ b½w þ 6w 6e ; w ¼ 0 (13) limp s s a s 1 6ws 2 þ w2s
Fig. 6. Diagram of perturbed bifurcations (sensitivity to initial conditions) of the partially imperfect system of Case Ia ðb ¼ 0:10Þ.
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while the other, which may be evaluated only numerically, is associated with non zero values of both generalized coordinates. Eq. (13) substantially describes the primary as well as the complementary (physically not accepted) equilibrium configurations of the partially imperfect system with only symmetric imperfections, which are in fact perturbations of the fundamental as well as the 1st secondary path of the perfect system. Evidently, the complementary path, standing for values of es, ws with different signs, possesses a limit point (local extremum) S1, while the fundamental path, for which es, ws have the same signs is monotonically rising.
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(a)
(b)
(c) Fig. 7. Projection of the equilibrium paths and critical points of the partially imperfect system of Case Ia, for es ¼ 0:025 and b ¼ 0:10.
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In order to investigate the perturbations of the bifurcations a thorough parametric study is carried out, by evaluating the critical points (and their stability) for a large range of values of es ð0:001pes p 0:50Þ. A partial symmetry breaking is revealed described in detail below. More specifically, the stable symmetric branching point C1 of the perfect system evolves to limit point S1 (as stated previously) and the system exhibits an independent dual-cusp catastrophe regardless of the value of es. Moreover, beyond this phenomenon, the unstable symmetric branching point C3 (on the 1st secondary path of the perfect system) gives birth to the also unstable symmetric bifurcation point C imp on the 3 monotonically rising fundamental path of the imperfect system. Its effect on the stability of the fundamental path is similar of the one of C3, since from the initial equilibrium configuration up to C imp the fundamental path is stable, becoming thereafter unstable. 3 Thus, a new dual-cusp singularity is revealed, initiated at C3, which is not independent, but connected to other dual-cusps, resulting from the perturbation of branching points C 03 and C2. Actually, for small values of jes j the complementary path exhibits (except S1) two unstable symmetric branching points, namely C 0imp and C imp 3 2 , which result from the perturbation of the also unstable symmetric bifurcation points C 03 and C2 of the 1st secondary and trivial path of the perfect system, respectively. As es increases absolutely, critical points C 0imp and C imp approach each other, and at jes j ¼ jecr s j ¼ 0:067 coincide and 3 2 then vanish. Thus, two more branches of interconnected perturbed bifurcations appear,
Fig. 8. 3D equilibrium paths and critical points of the partially imperfect system of Case Ia, for es ¼ 0:025 and b ¼ 0:10.
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starting from C2 and ending at C3, C 03 , to connect there with the dual-cusp due to the perturbation of C3. Consequently three interconnected dual cusps are present, a finding firstly reported in the literature [9,10]. As far as stability is concerned, only point C 0imp 3 changes, in conjunction with S1, the stability of the complementary path, while the presence of C imp does not affect the system’s overall stability. Of course, the presence of 2 imp critical points C imp and C 0imp is due to the intersection of the equilibrium branches on 2 , C3 3 which these exist with other secondary paths, the latter being the symmetry breaking perturbation of the third secondary path of the perfect system. In fact, the system besides the fundamental and complementary paths (on the [ws,l] configuration plane) possesses also two secondary branches on the configuration space [ws, wa, l], being curves having the
Fig. 9. As in Fig. 4, but for Case Ib.
Fig. 10. As in Fig. 5, but for Case Ib.
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above plane as a plane of symmetry. From these paths, the one is an open curve and exhibits two limit points S1 and S01 , being perturbations of branching points C4 and C 04 of the perfect system, respectively, creating two additional dual cusps on the configuration space. Their presence changes the stability of this particular path, which is unstable between them and unstable beyond them. Contrary, the 2nd secondary path, although possessing the same symmetry plane as the 1st one, is a closed curve connecting critical points C imp and C 0imp 2 3 , and is unstable throughout its length. All the preceding general findings are presented graphically throughout Figs. 4–8. More analytically, in Fig. 4a one may perceive the fundamental as well as the complementary equilibrium paths with their critical points for various values of the symmetric initial imperfection, while in Fig. 4b the stability of two characteristic such paths is presented (with the dashed line implying instability). In Fig. 5 all the critical points of the foregoing parametric study are depicted in 3D form, while the diagram of perturbed bifurcations on the [es,l] plane is shown in Fig. 6. As a paradigm, the case of a system with es ¼ 0:025 is dealt with in detail, and its corresponding equilibrium paths with all the critical points and stability branches are shown in 2D and 3D in Figs. 7(a,b,c) and 8, respectively. Case Ib: es ¼ 0; ea a0: For this sub-case also, Eqs. (12a,b) admit two type of solutions. The one related with zero symmetric deformation is given by the following analytical expression: 1 8ðwa ea Þ3 imp 3 ws ¼ 0; l2 ¼ þ 9b½wa þ 2wa 2ea (14) 6wa 2 þ 3w2a while the other, that may be evaluated only numerically, is associated with non zero values of both generalized coordinates, as in the previous sub-case. Eq. (14) stands for the fundamental as well as the complementary (physically not accepted) equilibrium configurations of the partially imperfect system with only antisymmetric imperfections. The fundamental paths correspond to ea, wa with equal signs, while the complementary ones, which always exhibit a local minimum—limit point S2, to ea, wa with opposite signs. Both these paths are the perturbations of the trivial as well as the 2nd secondary paths of the perfect system.
Fig. 11. As in Fig. 6, but for Case Ib.
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Fig. 12. As in Fig. 7, but for Case Ib with ea ¼ 0:025 and b ¼ 0:10.
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As already performed for Case Ia, a thorough parametric study is carried out, in order to investigate the perturbations of the bifurcations, by evaluating the critical points (and their stability) for a large range of values of ea ð0:001pea p 0:345Þ. The introduction of a single perturbation on the perfect system, namely of the antisymmetric initial imperfection, leads also to a partial symmetry breaking. More analytically, the unstable symmetric branching point C2 evolves into limit point S2 and thus the system exhibits an independent dual-cusp form singularity, regardless of the value of ea. The complementary equilibrium path, to which S2 belongs possesses also another critical point, the unstable symmetric branching point C 0imp 4 , being the perturbation of the also the unstable symmetric branching C 04 of the 2nd secondary path of the perfect system. The new dual cusp initiated from C 04 is not independent, but as it will be shown below, is connected with two more singularities of the same form, which are originated from critical points C1 and C4. Evidently, for small values of jea j the monotonically rising (and initially stable) fundamental equilibrium path of the imperfect system possesses two critical branching points, the stable symmetric bifurcation point C imp 1 , product of the perturbation of critical point C1 (of the corresponding perfect system), as well as the unstable bifurcation point C imp 4 , emerging from the perturbation of critical point C4 (of the 2nd secondary path of the perfect system). As ea increases absolutely, points C imp and C imp approach each other and 1 4
Fig. 13. As in Fig. 8, but for Case Ib with ea ¼ 0:025 and b ¼ 0:10.
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at jecr a j ¼ 0:101163 coincide and vanish, and hence the fundamental path does not have any critical point from this of value of jea j on. The two dual cusps resulting from C1 and C4 are consequently interconnected, and reach points C4, C 04 , where they meet the dual cusp due to C 04 . Hence, three dual cusp singularities are connected to each other, a very important new finding fist reported in the literature [9,10], similar to the one obtained for Case Ia. The presence of critical point C 0imp on the complementary path is the one separating it 4 between a stable and an unstable branch, while the initial position of limit point S2 on this specific path is not affecting its stability. Contrary, the branching points C imp and C imp 1 4 become (if existing) the limits between which the fundamental part is unstable and stable beyond them. According to the theorems of nonlinear stability, as also for Case Ia, the existence of bifurcation points on the primary and complementary paths, on the [wa,l] plane, is ought to the presence of two secondary paths in the configuration space, which have the above plane as a symmetry one, and are in fact symmetry braking perturbations of the 3rd secondary path of the perfect system. The 1st of these secondary paths passes across point C 0imp 4 , is an open curve, contains only unstable equilibria and has no other critical points. On the other hand, the 2nd secondary path is a closed curve connecting points C imp and 1
Fig. 14. Projections (a) and 3D representations (b) of the equilibrium paths of the fully imperfect system with es ¼ ea ¼ 0:025 (grey lines ¼ stability, black lines ¼ instability).
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Fig. 15. Example of the evolution of the critical points of the system due to imperfections ðb ¼ 0:10Þ.
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C imp and possesses also two limit point S2 and S02 , which are perturbations of points C3 4 and C 03 , respectively, creating thus two more independent dual cusps in the configuration space. These cusps start from the above points and asymptotically tend to that state of the imp system, for which at ea ¼ ecr and C imp coincide and then a the critical points C 1 4 disappear. The presence of the aforementioned limit point on this particular path affects its stability, since it is stable from C imp until these, and thereafter, until C imp 1 4 , unstable. All the above findings concerning the foregoing case are presented in graphical form throughout Figs. 9–13. Particularly, in Fig. 9a one may perceive the fundamental as well as the complementary equilibrium paths with their critical points for various values of the antisymmetric initial imperfection, while in Fig. 9b the stability of two characteristic such paths is presented (with the dashed line implying instability). Moreover, in Fig. 10 all the critical points of the foregoing parametric study are depicted in 3D form, while the
Fig. 16. Catastrophe Manifold Surface of the partially imperfect system of Case Ia: (a) of the first kind, (b) of the second kind.
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Fig. 17. Characteristic 3d views of the Catastrophe Manifolds of Fig. 16: (a) of the 1st kind, (b) of the 2nd kind.
diagram of perturbed bifurcations on the [ea,l] plane is shown in Fig. 11. Adopting lastly the value of ea ¼ 0:025 as an example, the equilibrium paths and their correspond stability branches along with all critical points are determined and shown in Figs. 12(a,b,c,) and 13 in 2D and 3D form respectively. Case II: Random imperfections: es ; ea a0. This last case of random imperfections, being in substance a double perturbation of the perfect system, cannot be systematically dealt with, within the Context of Nonlinear Stability and Bifurcation Theory, due to the strongly nonlinear nature of the governing equilibrium equations, but as it will be shown below, it may be comprehensively investigated using principles of the Theory of Catastrophes. Hence, the absence of analytical solutions for the various equilibrium configurations leads to their individual numerical evaluation (including critical points and stability) for each combinations of es, ea separately. For reason of comparison with the findings related to the perfect and partially imperfect system given earlier, the case of es ¼ ea ¼ 0:025 is studied and the corresponding equilibria, critical states as well as their overall stability is determined and presented graphically in Fig. 14. The system under this combination of parameters has been found to exhibit four (4) independent paths in the configuration space, possessing in total six (6) limit points, which are the evolutions—perturbations of the corresponding critical points of the partially imperfect system of either Cases Ia and Ib. These evolutions may be comprehensively perceived in the schematic diagram of Fig. 15, starting from the branching points of the perfect system. 2.2. Catastrophe theory approach 2.2.1. General considerations The partially imperfect system (Case Ia, Ib) is characterized by two main (active) coordinates, namely ws and wa, and also two control parameters, i.e. the loading l and the corresponding initial imperfection (either es or ea). In the sequel, its total potential does not
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belong to one of the seven elementary Catastrophes firstly introduced by Arnold [11]. Furthermore, the fully imperfect system, has two active coordinates and three control parameters but cannot be classified to one of the above 7 elementary singularities, since its potential differs tremendously from those of the hyperbolic or elliptic umbilic (which belong to these elementary catastrophes and have the same number of active variables and control parameters). Hence, no obvious classification or correlation exists and one must resort to the reconstruction of the Catastrophe Manifolds and Bifurcations Sets of the system, either partially or fully initially imperfect, and to the qualitative/quantitative study (to the most possible extent) of their salient features and properties. 2.2.2. Catastrophe manifolds (M) and bifurcations sets (Bs) According to the definitions and principles of the Theory of Catastrophes, the Catastrophe Manifolds, being the loci of equilibria, are in general hypersurfaces resulting from the equilibrium equations, while the Bifurcations Sets are generally also
Fig. 18. As in Fig. 16, but for Case Ib.
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hypersurfaces that require also the zeroing of the main stability determinant (Hessian) of the system. More specifically, for the partially imperfect system, employing advanced symbolic as well as graphical functions embedded in modern commercial mathematical software (Mathematica [6]) one may reconstruct two specific kinds of Catastrophe Manifold Surfaces, resulting from the elimination of one control parameter from Eqs. (12a,b), either the initial imperfection (es or ea) or the loading l. The resulting surfaces except their profound partial symmetry features, all exhibit the five basic characteristics of the dual-cusp Catastrophe, i.e. bi-modality, inaccessibility, sudden jumps, divergence and hysterisis. These properties are clearly depicted in Figs. 16–19, where various 3D views of the Catastrophe Manifolds of both aforementioned kinds is depicted. Moreover, in order to validate the results of the preceding Bifurcation Theory approach, the Catastrophe Manifold Surfaces of the 2nd kind (in the [ws, wa, imperfection] space) are redrawn, for both Cases Ia and Ib, the critical planes jes j and jea j are depicted in Fig. 20, while also the planes es ¼ 0:025 and ea ¼ 0:025 with all the critical points evaluated in the parametric studies are shown. From this particular figure the findings of the Bifurcation analysis are fully assessed, while the existence of independent and interconnected dual cusp singularities is comprehensively presented. In the same manner, for the partially imperfect system, the Bifurcation Sets are reconstructed as surfaces in the [ws, wa, imperfection] plane and presented in Fig. 21, revealing an extremely complicated pattern but nevertheless containing also the main characteristics of dual-cusps. This is not the case however for the Catastrophe Manifold of the fully imperfect system, since the matter concerns hypersufaces in a four dimensional space of the form Mðx; y; ws ; wa ; es ; ea Þ, and only projections of these on some mathematically convenient 3D space may be visualized. In doing these, for arbitrary values of es or ea , eliminating from the equilibrium equations the control parameter l, one may rather easily acquire
Fig. 19. As in Fig. 17, but for Case Ib.
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projections of the aforementioned hypersurfaces on either [ws,wa,ea] or [ws,wa,ws] spaces. The surfaces produced, as characteristically shown in Fig. 22, contain the main feature of full symmetry breaking and are related to multiple cusps, folds and limit points, being quite similar with the double-cusp singularity type (X9 as in [11]), which has been reported to be associated with several problems dealing with the buckling and postbuckling of rectangular plates [12,13], a finding validating the chosen simulation. Finally, the results of the Bifurcation Theory approach are fully validated, by reproducing sections of the Manifold projections for es ¼ 0:025 and ea ¼ 0:025 with the ea ¼ 0:025 and es ¼ 0:025 planes, respectively, as shown in Fig. 23.
Fig. 20. Catastrophe Manifold of the 2nd kind of the partially imperfect system. Left hand side (Case Ia): (a) cr surface with critical points, (b) surface with critical planes ecr s and es , (c) section with the es ¼ 0:025 plane (identical with Fig. 7a, Right-hand side (Case Ib): (d) surface with critical points, (e) surface with critical planes ecr a and ecr a , (c) section with the ea ¼ 0:025 plane (identical with Fig. 12a).
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3. Conclusions The most important conclusions drawn from this study are the following: The nonlinear 2-D.O.F. system dealt with in detail, exhibits an extraordinary postbuckling behavior, which for certain combinations of the parameters involved is related to remote postbuckling bifurcations, which connect/couple the secondary symmetric and antisymmetric equilibrium configurations and strongly affect their stability. Therefore, the danger of sudden, not expected transition (jump) from one to two mode buckling and visa versa is significant, for both perfect and partially initially imperfect system. This finding is of great importance for structural design purposes, while the possibility of the occurrence of such phenomena has been reported in the literature, for the buckling and postbuckling of rectangular plates, a fact validating the correct choice of the proposed model and the accuracy of its exact as well as approximate bifurcational analysis. The introduction of initial imperfections (partial perturbations) in the system (either only symmetric or only antisymmetric) leads to partial symmetry breaking, which
Fig. 21. Bifurcational Sets of the partially imperfect system for Case Ia (a) and Case Ib (b).
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Fig. 22. Various characteristic projections of the Catastrophe Manifold Hypersurface of the fully imperfect system ðb ¼ 0:10Þ.
gives birth to complicated bifurcations, related to independent and interconnected dual-cusp singularities, and the danger of mode jumping (although not usually related to imperfect systems) is visible. The fully nonlinear static buckling analysis produced results in full accordance with the ones gained from the Catastrophe Theory approach, the latter indicating singularities that cannot be classified in one of the seven elementary ones, although two active coordinates and two control parameters are involved. Finally, introducing an arbitrary full initial imperfection in the system, considered either as an overall perturbation of the perfect one or as a single perturbation of the partially imperfect one, leads to the evolution of all critical points to limit point ones, combined with total symmetry breaking. Through a characteristic numerical application, the full accordance of the findings of both Bifurcational and Catastrophe Theory approaches was demonstrated, the latter revealing the existence of a complicated higher order two-mode singularity, containing the main features of the double-cusp, which has been reported being related with the buckling of plates. Hence, once again the accuracy and validity of the proposed simulation is confirmed.
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Fig. 23. Projections of the Catastrophe Manifold Hypersurface of the fully imperfect system. Left-hand side (a,b,c): On the [ws,wa,ea] space for es ¼ 0:025 and section with the ea ¼ 0:025 plane, Right-hand side (d,e,f): On the [ws,wa,es] space for ea ¼ 0:025 and section with the es ¼ 0:025 plane—full accordance with Fig. 14c,d.
References [1] V. Gioncu, M. Ivan, Theory of critical and postcritical behavior of elastic structures, Editura Academiei Republicii Socialiste Romania, 1984. [2] W.J. Supple, A.H. Chilver, Elastic post-buckling of compressed rectangular plates, in: A.H. Chilver (Ed.), Thin-Walled Structures, Chatto & Windus, London, 1967. [3] H. Troger, A. Steindl, Nonlinear Stability and Bifurcation Theory: An Introduction for Engineers and Scientists, Springer, Wien/NY, 1991. [4] R. Gilmore, Catastrophe Theory for Scientists and Engineers, Dover, UK, 1993. [5] T. Poston, I. Stewart, Catastrophe Theory and its Applications, Pitman, London, 1978. [6] T.B. Bahder, Mathematica for Scientists and Engineers, Addison-Wesley, New York, 1995. [7] T. Nakamura, K. Uetani, The secondary buckling and post-secondary buckling behaviors of rectangular plates, Int. J. Mech. Sci. 21 (1979) 256–286. [8] P.R. Everall, G.W. Hunt, Anrold tongue predictions of secondary buckling in thin elastic plates, J. Mech. Phys. Solids 47 (1999) 2187–2206.
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[9] D.S. Sophianopoulos, L. Argyropoulou, E. Evagorou, V. Marinithou, Secondary bifurcations in postrcritical domains of a two-degree-of-freedom nonlinear system with elements of catastrophe theory, in: A.N. Kounadis, C. Povidakis, G. Exadaktylos (Eds.), Proceedings of the Seventh National Congress on Mechanics, Chania, Crete, Greece, June 24–26, 2004, Technical University of Crete, vol. I, pp. 244—249. [10] D.S. Sophianopoulos, Bifurcations and Catastrophes of two-degree-of-freedom nonlinear model simulating the buckling and postbuckling of rectangular plates, in: Proceedings of the First International Conference on Modeling, Simulation and Applied Optimization, Sharjah, U.A.E., February 1–3, 2005 (Paper ID: 132, Proc. in CD-ROM, ICMSAO 2005, ISBN: 9948-427-00-9). [11] V.I. Arnold, Catastrophe Theory, third ed., Springer, Berlin, 1992. [12] J.M.T. Thompson, Instabilities and Catastrophes in Science and Engineering, Wiley, New York, 1982. [13] J.M.T. Thompson, G.W. Hunt, Elastic Instability Phenomena, Wiley, Chichester, 1984.