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Arnold tongue predictions of secondary buckling in thin elastic plates
J. Mech. Phys. Solids 47 (1999) 2187±2206
Arnold tongue predictions of secondary buckling in thin elastic plates P.R. Everall*, G.W. Hunt Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK Received 24 August 1998
Abstract The stability of post-buckled states for simply-supported ¯at elastic plates under compression is investigated for a range of in-plane boundary conditions. The von KaÂrmaÂn plate equations are reduced to a series of ODEs which are solved numerically under parametric variation of both load and length. Results are checked against full numerical solutions of the PDEs, and comparison with a modal analysis highlights the dominant passive contaminations. The nondimensional amplitude at secondary bifurcation, for any combination of modes and all plate lengths, is presented in a concise form using the parameter space of Arnold tongues. This demonstrates that compound bifurcation represents a worst case for post-buckling reserve, and that long plates have inherently more such reserve than short plates. It is also shown that stiening the boundaries against inplane movement is destabilizing, in that it induces mode jumping at secondary bifurcation to occur at an earlier stage in the post-buckling regime. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: B. Structures; C. Energy methods; Numerical algorithms; Mode jumping
1. Introduction Thin elastic plates have many applications which exploit their stable postbuckling response in compression. The aim of this paper is to investigate the limit of this stability for simply-supported ¯at plates subject to various in-plane boundary conditions. The post-buckling behaviour of such plates is sometimes * Corresponding author. 0022-5096/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 5 0 9 6 ( 9 9 ) 0 0 0 0 8 - 3
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explored using modal approximations (see, for example, Supple (1970), Uemura and Byon (1977) and Maaskant and Roorda (1992)). This has been successfully used to predict the type of coupled solution path (Supple, 1967), and the possibility of mode jumping, for plates with various boundary conditions. Results have been compared with experiments for one case (Sharman and Humpherson, 1968), although in general the limitations of the method have not been fully explored. Schaeer and Golubitsky (1979) use an analytical method to demonstrate mode jumping in straight-edged plates clamped in bending. Alternatively, Gervais et al. (1997) have shown the possibility of mode jumping in partially and fully clamped plates using a ®nite element method, but this turns out to be a time consuming process over a large range of aspect ratios and for dierent boundary conditions. Finite element and experimental results for a clamped plate have also been compared by Uemura and Byon (1978). This paper compares modal and numerical methods with a third approach, the reduction of the coupled partial dierential equations (PDEs) by the Galerkin procedure to a series of ordinary dierential equations (ODEs), which are solved using the boundary value solver AUTO (Doedel et al., 1995). This is successful because the mode interaction which leads to the loss of stability maintains a transverse pro®le that is similar at all positions along the length. Results are compared with full numerical solutions. Limitations of Rayleigh±Ritz (modal) analysis are also explored by comparing predictions of the secondary bifurcation points with those given by AUTO. The paper presents its ®ndings in the form of Arnold tongue plots (Arnold, 1965), borrowed from nonlinear dynamical systems theory (Thompson and Stewart, 1986). Previously presented for the ODE problem of a strut supported by a stabilizing elastic medium (Hunt and Everall, 1999), these summarize the coupled behaviour in a succinct manner.
2. Theory The post-buckling of a thin plate is described by the von KaÂrmaÂn largede¯ection equations, which in nondimensional form can be written r4 w wxx fyy ÿ p2 Px ÿ 2wxy fxy wyy fxx ÿ p2 Py ,
1
r4 f w2xy ÿ wxx wyy ,
2
where subscripts denote partial dierentiation with respect to the variable. Px and Py are the average applied loads per unit of length in the x and y directions, respectively, w is the vertical displacement and f is a stress function describing the change from the uniformly compressed state, de®ned by fyy
txy sy sx , fxy ÿ , fxx , E E E
3
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where stresses sx etc. again represent the change from uniform compression. The ®rst is an equilibrium and the second a compatibility equation. 2.1. Boundary conditions A simply-supported plate of unit width and length L is considered with three dierent sets of in-plane boundary conditions. For all three cases, shown in Fig. 1, there are no out-of-plane (z ) de¯ections or bending moments at the plate edges. A plate with no applied or reactive stresses at the boundaries, apart from the load Px, is shown in Fig. 1(a). All edges are free to move in the plane of the plate. In this case no secondary bifurcations occur on the natural loading path (Supple, 1970), mode jumping is absent and hence further analysis is not pursued. However, mode jumping has been observed when the edges are held straight; Fig. 1(b) shows that to achieve this, although Py=0, the stresses at the plate boundaries are no longer uniform. The variation from the uniformly compressed state is described by f, which has the following constraints: 1 fyy dy fxy 0 at x 0,L, 0
L 0
fxx dx fxy 0 at
y 0,1:
4
Fig. 1. Simply supported plate with: (a) edges free to move; (b) edges held straight; (c) long edges clamped in-plane. Broken lines show unloaded plate. Solid lines show post-buckled shape.
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For the plate that is clamped in-plane (Fig. 1(c)), such that the long boundaries are restrained from moving in the transverse direction but remain free to move longitudinally, the average transverse applied stress is non-zero. Before buckling it is clear that Py=nPx (where n is Poisson's ratio); however this no longer holds in the post-buckling regime, where 1 1 2 2 fxx ÿ nfyy ÿ wy dy: 5 p Py ÿ nPx 2 0 This is an integral condition across the width of the plate, and it is used directly in all the following numerical solutions to determine Py. Again, the boundary conditions for f are de®ned by (4). In the remainder of this paper we investigate the behaviour of plates subjected to the boundary conditions shown in Fig. 1(b) and (c). These will be termed `straight-edged' and `clamped in-plane' boundaries, respectively. In addition we will vary the single loading parameter, P 0Px. 2.2. Critical loads The critical loads can be found by linearizing (1) and (5), to give r4 w ÿwxx p2 P ÿ wyy np2 P
6
for the clamped in-plane boundary conditions of Fig. 1(c). Substituting w=A Lpx sin py, leads to the characteristic equation, L4 L2 P ÿ 2 1 ÿ nP 0: When 4 ÿ 4n < P < 1/n, L2 is real, s P ÿ 2 P ÿ 22 2 nP ÿ 1, L2 ÿ 2 4 leading to two imaginary eigenvalues de®ned by v s u u tP ÿ 2 P ÿ 22 ÿ nP ÿ 1, l1 2 4 v s u u tP ÿ 2 P ÿ 22 l2 nP ÿ 1: 4 2
7
8
9
(The two negative eigenvalues can be dropped because of the reversibility in x.) Over a ®nite length L however, to satisfy the simply-supported boundary conditions, the de¯ection must be an integer number of half-waves, n,
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w A sin lpx sin py Qn sin
npx sin py, L
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10
and therefore l=n/L (where l is either l1 or l2). Substituting this value into (7) gives the critical loads for buckling into n half-waves at all plate lengths, Pn
n2 L2 2 : L2 n2 nL2
11
The values of l and Pn for the straight-edged boundary conditions of Fig. 1(b) can be found by substituting n=0 into (9) and (11), respectively. The variations in critical load for the ®rst four mode numbers are shown in Fig. 2(a) over a range of plate lengths. Critical loads for the clamped in-plane plate are lower than those for the straight-edged plate, with their minima occurring at relatively greater lengths. This is due to pre-buckled compression across the width, causing it to buckle at a load lower than when such compression is absent. At discrete plate lengths, the critical loads for dierent modes coincide at compound bifurcation points. Fig. 2(b) shows the variation of l1/l2 with load for the two cases. For the straight-edged plate, l1/l2 4 0 as P 4 1 and therefore compound bifurcation points can occur for all P > 4. However, if the edges are clamped in-plane, they are restricted to the range 4 ÿ 4n < P < 1/n; this is apparent from Fig. 2(a), where the critical loads for all mode numbers fall into a narrow band as the plate length increases. 2.3. A typical Arnold tongue Fig. 3(a) illustrates schematically the general form of the equilibrium solution paths for the following two modes (Supple, 1967),
Fig. 2. (a) Critical loads and (b) variation of l1/l2 with load. Solid line: straight-edged; broken line: clamped in-plane with n=0.3.
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apx bpx Qb sin w Qa sin L L
sin py,
12
where a < b, at a length just to the left of the corresponding compound bifurcation point of Fig. 2(a), where the curves of Pa and Pb cross. Bifurcation from the ¯at state occurs at the critical load Pa, whereupon the system buckles into a half-waves and follows the stable uncoupled path (Qb=0). The uncoupled path with b half-waves (Qa=0) intersects the fundamental path at Pb>Pa, and is therefore initially unstable with respect to Qa. Sa and Sb, respectively mark the positions where a coupled solution, for which Qa and Qb are both non-zero, intersects the lower and upper uncoupled paths. The coupled path projects onto the base plane as a closed loop as shown. The lower uncoupled path thus loses stability at Sa, where under dead loading conditions a dynamic jump takes place to the upper uncoupled path (Qa=0), which has stabilized at Sb. This is a mode jump from a to b half-waves. The position at which secondary buckling occurs for all plate lengths has been plotted as a function of load by Nakamura and Uetani (1979). This indicates the lowest secondary buckling load for any given plate length, but not the buckle pattern after mode jumping. In contrast, an Arnold tongues plot (Arnold, 1965) describes the secondary bifurcation positions for any two modes, at all plate lengths (Hunt and Everall, 1999). Arnold tongues were ®rst used to describe the behaviour of a second-order nonlinear forced oscillator (Thompson and Stewart, 1986). In a typical such system two frequencies are of interestÐa natural frequency and that of the forcing function. The tongues describe cusp-shaped loci which, in the space of the ratio of these frequencies and the amplitude of forcing, mark the limit of mode locking. For the plate, the ratio of the two natural `frequencies' is plotted against nondimensional amplitude (Fig. 3(b)), with the cusps marking the amplitudes of Sa and Sb for all plate lengths, and the ratio of
Fig. 3. (a) Equilibrium paths. Stable paths are shown as solid lines and unstable paths as broken lines. (b) Tongue for straight-edged plate, showing interaction between 2 and 3 half-waves.
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the interacting mode numbers being given by the point where the cusp meets the l1/l2 axis. For example, the tongue depicting the interaction between a = 2 and b = 3 for the straight-edged plate is plotted in Fig. 3(b). The values of both l1 and l2 for each of the two modes of interest are ®xed by its critical load, Pa or Pb (11). While one of these l values must be linked directly to an integer (a or b ), the other in general will not; typically therefore l1/l2 is irrational. Only when the length is such that two critical loads coincide will both l values be linked to integers, and l1/l2 be rational. The compound bifurcation point where Pa=Pb is thus marked by a rational position on the l1/l2 axis, in this case at l1/l2=2/3. Note also that the minimum position of each curve of Fig. 2(a) has l1=l2=1 and thus appears on the Arnold tongue plot at the right-hand margin. As we move to the left of the compound bifurcation point in Fig. 2(a), in this example at the intersection of the curves n = a=2 and n = b=3, Pb increases and hence the l1/l2 related to it decreases (Fig. 2(b)), while at the same time Sb emerges from the compound bifurcation point and moves up the uncoupled path. This is traced by the left-hand arm of the Arnold tongue, shown as a solid line in Fig. 3(b). The vertical axis gives the nondimensional amplitude Qb at the secondary bifurcation Sb. Of more practical interest is the position of Sa, represented by the right-hand arm of the cusp. Again to the left of the relevant intersection of Fig. 2(a), l1/l2 increases as Pa reduces, and the position of Sa is given by the solid line to the right of l1/l2=2/3. However, unlike Sb, Sa fails to pull in to the compound bifurcation point as Pa and Pb converge. Instead it remains remote from the fundamental path, and vanishes only when it collides with a further secondary bifurcation on the same uncoupled path, which is found to emerge from the compound bifurcation point in the region to the right of the n = 2 and n = 3 intersection of Fig. 2(a). (This complex mechanism is examined in detail in Hunt and Everall (1999).) These two secondary bifurcations are represented in Fig. 3(b) by the broken line in the right-hand arm of the tongue which initially veers to the left; as in this region Pa>Pb, they cannot be reached in a normal loading sequence. There is thus a certain threshold value of amplitude of Sa below which mode jumping is absent, which is picked out at l1/l2=a/b = 2/3, where the broken line changes to solid. Hence mode jumping only occurs at positions to the left of the relevant intersection of Fig. 2(a), and always involves a reduction in the wavelength of the buckle pattern, from a to b>a half-waves. p For the p straight-edged plate when a = 2 and b = 3, Pa=Pb occurs at L 6, so for L< 6 2:4 say, the coupled behaviour is similar to that shown in Fig. 3(a). The values of l1/l2 at Pa and Pb are now 0.694 and 0.64, respectively, shown by the vertical dotted lines in Fig. 3(b). The nondimensional amplitudes of the secondary bifurcation points are given by the intersection of these lines with the two arms of the cusp, in this example Qa 1 4.5 and Qb 1 0.4. No practicallyrelevant secondary bifurcation positions with Qa < 4.09 exist on the lower uncoupled path. In dimensional terms, where
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p 12 1 ÿ n2 qa , Qa t
13
this translates into an actual amplitude of qa 1 1.24t, where t is the plate thickness.
3. Analytical approaches 3.1. Numerical solution of the PDEs There are no simple exact solutions of the PDEs (1) and (2), so a classical ®nite dierence method is used to give a series of nonlinear algebraic equations. These equations are solved for w and f using an iterative method, the current solution values being substituted back into the algebraic equations and used to give an improved solution. This procedure is repeated until the change is less than a desired tolerance. Although this technique is simple to implement, it takes a long time to converge and does not locate secondary bifurcation points. A more sophisticated ®nite element program has been written (Gervais et al., 1997) which tracks the post-buckled paths and identi®es the secondary bifurcation points. However, this method remains time consuming, especially for large aspect ratios where many grid points are required to maintain accuracy. 3.2. Reduction to ODEs AUTO (Doedel et al., 1995) is a boundary value solver which is able to track solution paths and log secondary bifurcation points for ODEs, with much quicker solution times than for the PDEs. By assuming that the form of the de¯ection across the plate ( y ) remains the same at all positions along the length (see also Lord et al. (1997)), we can reduce the PDEs to a series of ODEs in x. The following functions w x,y w1 x sin py, f x,y f1 x f2 x cos 2py,
14
satisfy the boundary conditions on the long edges de®ned in (4) and (5). These are substituted into (1) and (2), and the Galerkin procedure is then used to produce a series of ODEs, which with the appropriate boundary conditions, are solved in AUTO. Unlike a Rayleigh±Ritz analysis which uses a limited number of modes, this method allows the interaction between all periodic forms along the length to be included. It has been shown (Nakamura and Uetani, 1979) that the eect of the ®rst transverse harmonic can be signi®cant at large amplitudes. This is included by using two displacement functions,
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w x,y w1 x sin py w2 x sin 3py, f x,y f1 x f2 x cos 2py f3 x cos 4py f4 x cos 6py:
15
However, as this doubles the number of ODEs that need to be solved, solution times will increase. The solutions of (14) and (15) are validated with the full numerical solution of the PDEs described above at discrete loads and aspect ratios as described in Section 4.1. 3.3. Rayleigh±Ritz The total potential energy, in nondimensional form, for an elastic plate of unit width and length L can be written (Lord et al., 1997), V
1 2
1 L
1 2
0 0
r2 w2 ÿ 2 1 ÿ n wxx wyy ÿ w2xy dxdy
1 L 0 0
1 ÿ p2 P 2
r2 f2 ÿ 2 1 n fxx fyy ÿ f2xy dx dy
1 L 0 0
16
w2x dx dy:
Sinusoidal functions along the length of the plate and the transverse functions given in (14) satisfy the boundary conditions (4) and (5). The stress function f is found by substituting these functions for w into (2) and integrating directly; compatibility is thus automatically satis®ed and a Lagrange multiplier is not required. These expressions for w and f are substituted into (16), leading to the energy function, V
1 XXXX 1X 0 Vabcd Qa Qb Qc Qd V aa P ÿ P a Q2a , 24 a b c d 2 a
17
where a, b, etc. are positive integers. Like the strut on a hardening foundation (Hunt and Everall, 1999), the energy function carries a fundamental symmetry in that it contains only even powers of w. Zero and non-zero Vabcd terms thus occur for the same combinations of a, b, c and d as for the strut, although the non-zero coecients are more complex (see Appendix A). Equilibrium solutions are given where all the ®rst variations of (17), @V , @Qa
@V , @Qb
18
etc., are simultaneously zero. Hence a ®rst order approximation of the load for the straight-edged plate is given by
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P Pa
a4 L4 Q2a : 16a2 L2
19
This modal approach can be used to predict the positions of the secondary bifurcations (Sa and Sb ) using a ®nite number of modes. The simplest approximation uses two modes (12), for which Supple (1967) shows that the amplitude at Sa is given by Qa
3V 0
6V 0 aa V 0 bb P b ÿ P a : 0 aa Vaabb ÿ V bb Vaaaa
20
However, previous work (Hunt and Everall, 1999) has shown that this is only accurate for small amplitudes. The agreement at large amplitudes can be improved by including other modes in the approximation; these are denoted by Greek letters and called passive modes because, in contrast to the `active' modes a and b, they have minimal eect close to the primary bifurcation points. When one passive mode is included, say a=2a ÿ b, the secondary bifurcation occurs where the determinant (Thompson and Hunt, 1973), 2 @ V=@ Q2 b @ 2 V=@ Qb @Qa
@ 2 V=@ Qb @ Qa , @ 2 V=@ Q2a
21
evaluated on the lower uncoupled path, vanishes. This analysis is essentially equivalent to a general Lyapunov±Schmidt reduction (Golubitsky and Schaeer, 1984). The approximation for the uncoupled path in Qa can be improved by including either the longitudinal, apx 3apx Qb sin sin py, w Qa sin L L
22
or both the longitudinal and transverse harmonics, apx 3apx Qb sin sin py w Qa sin L L apx 3apx Qd sin sin 3py, Qg sin L L
23
although (17) and (21) both increase in complexity. Again, the solutions can be validated against the numerical solution, and the secondary bifurcation positions given by this analysis compared with those from AUTO. Therefore, unlike other work (Supple, 1970; Uemura and Byon, 1977; Maaskant and Roorda, 1992), the dominant harmonics and passive modes can be identi®ed and the limitations of this approach assessed.
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4. Results 4.1. Validation 4.1.1. Uncoupled paths The solutions given by the three methods described above are compared by calculating the total energy (16) of a square plate (L = 1) at dierent loads. Fig. 4(a) shows the percentage dierence between the energy of the numerical solution and values given by the AUTO and Rayleigh±Ritz methods, for the straight-edged plate. The solution of (14) by AUTO has an error of approximately 3% at 3Pn, but with inclusion of the transverse harmonic (15) this reduces to 0.1%, giving excellent agreement with the numerical solution. Solutions of a single mode Rayleigh±Ritz analysis are poor, but addition of the ®rst harmonic (22) gives solutions very similar to the same approximation in AUTO. Solutions of the four mode approximation (23) are hard to distinguish from both the numerical solution and those arising from the same transverse variation in AUTO. This implies that uncoupled paths can be accurately modelled by these four modes. Fig. 4(b) shows similar trends for a plate with the clamped in-plane boundary conditions. 4.1.2. Secondary bifurcations Like the numerical solution of the PDEs, locating the secondary bifurcations using AUTO for long plates is time consuming. Therefore AUTO was also used to validate the predictions given by the Rayleigh±Ritz approximation, which gives solutions more quickly. The dierences between AUTO and three modal approximations are summarized by the tongue describing the interaction between two and three half-waves, shown in Fig. 5 for the two sets of boundary conditions. The simplest approximation plotted includes a, b and one passive
Fig. 4. Percentage errors in energy (16) between numerical solution and: Rayleigh±Ritz solution of (17) with (i) 1, (ii) 2 and (iii) 4 modes (broken line); AUTO solution of (iv) (14) and (v) (15) (solid line). (a) Straight-edged; (b) clamped in-plane.
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contamination, a=2a ÿ b; Fig. 5(b) shows this only gives good agreement for Qn < 1. This is expected, as the single mode approximation for the uncoupled path (curve (i) of Fig. 4(b)) is poor. When the eects of the ®rst harmonic (22) and two other passive modes (2a+b and 4a ÿ b ) are introduced, acceptable agreement extends to Qn 1 3 (Fig. 5(a)). For the clamped in-plane boundary conditions, secondary buckling occurs at relatively low loads where the error using this approximation is less than 1% (curve (ii) of Fig. 4(b)); Fig. 5(b) shows that this is indistinguishable from the numerical results. This approximation also gives accurate predictions for Sb, as the amplitudes of the secondary bifurcations on the upper uncoupled path are also relatively small. Fig. 5(a) shows that inclusion of the transverse harmonic (23) in conjunction with the same passive modes (3a, 2a ÿ b, 2a+b and 4a ÿ b ), gives accurate predictions when Qn < 6. However, this gives a very lengthy energy function (17) and a 64 element stability matrix (21); both are dicult to manipulate and time consuming to solve. Therefore, for the straight-edged plate, a modal approximation of sucient accuracy is not signi®cantly quicker than AUTO. It also highlights the increase in the number of passive modes as Sa moves away from the primary bifurcation point. 4.2. Arnold tongues The Arnold tongues for plates with straight-edged and clamped in-plane boundary conditions are plotted in Fig. 6(a) and (b), respectively, using the solutions of (15) given by AUTO. Most secondary bifurcations occur on uncoupled paths which are already unstable with respect to another mode and therefore have no practical signi®cance. For clarity, only the ®rst four tongues which mark secondary bifurcations associated with the initial loss of stability are shown. As described in Section 2.3, the tongues always meet the l1/l2 axis at rational positions, giving the ratio between the number of half-waves before and after mode jumping, a/b. For the clamped in-plane plate (Fig. 6(b)), the fact that
Fig. 5. Comparison between numerically (solid line) and modally (broken line) produced tongues with (i) 3, (ii) 6, (iii) 12 modes. (a) Straight-edged; (b) clamped in-plane.
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Fig. 6. Arnold tongues produced from AUTO results (a) straight-edged; (b) clamped in-plane.
the tongues do not cross suggests that the case of practical relevance is always b=a + 1. However, for the straight-edged plate (Fig. 6(a)), some of the righthand arms do cross, indicating that mode jumping may take place with, say, the mode b=a + 2. All the illustrated tongues curve back on themselves, as seen in Fig. 3, and so the smallest amplitude of secondary buckling is given when l1/l2=a/b (and hence Pa=Pb ) for each combination of modes; fully-coincident buckling thus reduces the safety margin in the post-buckling range, as might be expected. The locus of these points, shown as a dotted line, is thus a safety envelope, inside which no secondary buckling can occur. Tongues with low values of l1/l2, for example 1/4, are outside the safety envelope, with the right hand arm moving initially to the right, so mode jumping could occur at vanishingly-small amplitudes just above Pa (Hunt and Everall, 1999). However, these interactions occur on unstable uncoupled paths, and are not practically relevant. What is perhaps more surprising is that for each of the plots of Fig. 6, the safety envelope increases in the Qn sense as l1/l2 4 1. This implies that, as the plate gets longer and the interacting wavenumbers get relatively closer together, secondary bifurcation is increasingly delayed. This perhaps seems counterintuitive, as it might be thought that the interactions present in shorter plates, where the boundary conditions place considerable restriction on the choice of wave, would always be available to the less restricted longer plate. That this is not the case is attributable to sympathetic mode locking. Consider for example the interaction between a = 4 and b = 5, for the plate that is clamped in-plane. The most severe interaction takes place when l1/l2=4/5 on the Arnold tongue picture of Fig. 6(b). However it is not the case that this particular interaction would occur in a plate of exactly double the length. The critical load plot of Fig. 2(a) reveals that at this length the plate would buckle into a = 9 half-waves, and subsequently interact with b = 10. The worst-case interaction for this latter problem is now at l1/l2=9/10 on the Arnold tongue
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picture, with the safety envelope showing that secondary bifurcation is inevitably further delayed. Note that the Arnold tongues plot indicates the positions of secondary bifurcations, whereas stability may also be lost at a limit point, either before or after secondary buckling. The complete picture of the coupled paths, which may be distorted greatly by passive eects, is now explored using AUTO. 4.3. Passive mode eects 4.3.1. Limit point instability Even at relatively small amplitudes the eect of the passive modes can be signi®cant; when the primary mode is a single half-wave for example, the amplitude of the ®rst passive harmonic (b=3a = 3) also becomes large when Qa is large. Fig. 7 shows the contaminating mechanism by which stability is lost to this harmonic, before the secondary bifurcation marking interaction with a two halfwave solution is encountered. The contamination is due to Vaaab, which, being negative (see Appendix A), is destabilizing in nature, whereas Vaabb is positive and stabilizing. Thus stability is lost at a limit point which is not picked out by the Arnold tongues plot, and accounts for the absence of a tongue at l1/l2=1/2 for both sets of boundary conditions. 4.3.2. Stable secondary bifurcation For the clamped in-plane plate, secondary buckling from a = 2 to b = 3 occurs before the uncoupled path is destabilized by the passive harmonic b=3a = 6. However, Fig. 8(a) shows that at Sa the coupled path is initially stable. Mode jumping is then delayed until a limit point where P > PSa; again, in this instance, the tongue does not show the amplitude where stability is lost. For slightly shorter lengths (Fig. 8(b)), the isolated solution path of Fig. 8(a) merges with the coupled path. This gives a very complex coupled path, with the amplitudes of some passive modes being greater than Qa and Qb, highlighting the increasing eect of passive
Fig. 7. Eect of ®rst harmonic at Pa=Pb for the clamped in-plane plate. Stable paths are shown as solid lines and unstable paths as broken lines.
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modes with displacement away from both primary and secondary bifurcation points. Stability is now lost at a higher load than might be expected from Fig. 8(a), with the path restabilizing before a jump to the 3 half-wave solution occurs. 4.3.3. Unstable secondary bifurcation As L increases, the coupled path moves from stable to unstable. Fig. 9(a) shows that mode jumping occurs at the secondary bifurcation point when a = 4. For slightly shorter lengths (Fig. 9(b)) the coupled path again merges with the isolated solution path, and although it restabilizes the load reached is less than PSa, implying that under dead loading conditions the buckle pattern will change instantaneously. Therefore the tongues mark the limit of stability when a r4. 4.3.4. Crossing of tongues The amplitudes of Sa for the straight-edged plate are much larger than for clamped in-plane boundary conditions, and hence the eect of the passive modes will be greater and the coupled behaviour more complex. The tongue of l1/l2= 3/5 crosses that of l1/l2=3/4, showing that when a = 3 interaction can occur with the mode b=a + 2=5; this coupled path contains only odd modes which have stronger coupling (V3355>V3344 for example). When b=a + 1, the coupled path is symmetrical with respect to both Qa and Qb. However, when a and b are both odd this double-symmetry is broken; ¯ipping one mode through while keeping the other unchanged generates a dierent shape. This is re¯ected in the plot of Fig. 10, in which the coupled path does not project onto the base plane as a closed loop. 5. Conclusions The paper explores the eect of the in-plane boundary conditions on the postbuckled response of a simply-supported ¯at plate. As the boundaries are
Fig. 8. Jump from 2 to 3 waves for the clamped in-plane plate, (a) L = 3.80, (b) L = 3.76. Stable paths are shown as solid lines and unstable paths as broken lines.
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Fig. 9. Jump from 4 to 5 waves for the clamped in-plane plate, (a) L = 6.70; (b) L = 6.60. Stable paths are shown as solid lines and unstable paths as broken lines.
increasingly constrained against in-plane movement, the amplitude, and hence the load, at which secondary bifurcation occurs reduces: the unrestrained plate of Fig. 1(a) is unconditionally stable; for most lengths the straight-edged plate loses stability at P > 2Pn; the plate with its edges clamped in-plane exhibits little by comparison in the way of post-buckled strength. The most useful results are obtained by reducing the von KaÂrmaÂn equations to a set of ODEs which are solved numerically using AUTO, hence including the destabilizing eects of all longitudinal harmonics and passive modes. Validation with the full numerical solution of the PDEs highlights the eect of transverse harmonics, especially at large amplitudes. Comparisons with Rayleigh±Ritz results show that passive modes can signi®cantly aect the amplitude of secondary buckling, so the coupled behaviour can be complex. The results are presented using the parameter space of Arnold tongues. This plot shows the amplitude at which secondary buckling occurs for all plate lengths,
Fig. 10. Jump from 3 to 5 waves for the straight-edged plate. Stable paths are shown as solid lines and unstable paths as broken lines.
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with the point where the tongue meets the l1/l2 axis giving the ratio between the wavelengths before and after mode jumping. The safety envelope marks the smallest amplitude at which secondary buckling can occurÐit is noted with interest that this amplitude increases as L 4 1. However, the tongue picture does not indicate the behaviour or stability of the coupled path; in general for short plates the coupled path is stable, with mode jumping only occurring at the secondary bifurcation point for larger aspect ratios. In most practical applications ¯at plates are not simply-supported, and therefore it might be interesting to extend this work to encompass other than simply-supported boundaries. The post-buckled wavelength can now modulate with length, and hence it may not be possible to present the results directly in the form of Arnold tongues.
Acknowledgements The authors are grateful to Gabriel Lord for supplying a working version of his Galerkin reduction. This work has been ®nanced by an Engineering and Physical Sciences Research Council (EPSRC) research studentship.
Appendix A. Energy coecients An algebraic manipulation package was used to calculate the energy coecients in (16). The non-zero values for the straight-edged plate are, 3 4 a4 A1 Vaaaa p L 1 4 , L 32
Vaabb
" # 1 4 a2 b2 a ÿ b4 a b4 p L 4 4 , L 32 a b2 4L2 2 a ÿ b2 4L2 2
Vaaab ÿ
3 4 p L 32
for
b 3a,
A2
A3
Vaabc
" # 1 4 a b2 3a ÿ b2 p L 3 32 a ÿ b2 4L2 2
Vaabc
" # 1 4 a ÿ b2 3a b2 ÿ p L 3 for 32 a b2 4L2 2
for
c 2a ÿ b,
c 2a b,
A4
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Vabcd
" 1 4 a ÿ b2 a b 2c2 p L 6 64 a b2 4L2 2
# a ÿ c2 a 2b c2 b ÿ c2 2a b c2 for d ÿa ÿ b ÿ c, a c2 4L2 2 b c2 4L2 2
Vabcd
" 1 4 a ÿ b2 a b ÿ 2c2 ÿ p L 6 64 a b2 4L2 2
# a c2 a 2b ÿ c2 b c2 2a b ÿ c2 for d ÿa ÿ b c, a ÿ c2 4L2 2 b ÿ c2 4L2 2
Vabcd
" 1 4 a b2 a ÿ b 2c2 ÿ p L 6 64 a ÿ b2 4L2 2
# a ÿ c2 a ÿ 2b c2 b c2 2a ÿ b c2 for d ÿa b ÿ c, a c2 4L2 2 b ÿ c2 4L2 2
Vabcd
" 1 4 a b2 a ÿ b ÿ 2c2 p L 6 64 a ÿ b2 4L2 2
# a c2 a ÿ 2b ÿ c2 b ÿ c2 2a ÿ b ÿ c2 for d ÿa b c, a ÿ c2 4L2 2 b c2 4L2 2
Vabcd
" 1 4 a b2 a ÿ b ÿ 2c2 ÿ p L 6 64 a ÿ b2 4L2 2
# a c2 a ÿ 2b ÿ c2 b ÿ c2 2a ÿ b ÿ c2 for a ÿ c2 4L2 2 b c2 4L2 2
Vabcd
d a ÿ b ÿ c,
" 1 4 a b2 a ÿ b 2c2 p L 6 64 a ÿ b2 4L2 2
# a ÿ c2 a ÿ 2b c2 b c2 2a ÿ b c2 for d a ÿ b c, a c2 4L2 2 b ÿ c2 4L2 2
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2205
"
Vabcd
1 4 a ÿ b2 a b ÿ 2c2 p L 6 64 a b2 4L2 2
# a c2 a 2b ÿ c2 b c2 2a b ÿ c2 for d a b ÿ c, a ÿ c2 4L2 2 b ÿ c2 4L2 2
Vabcd
" 1 4 a ÿ b2 a b 2c2 ÿ p L 6 64 a b2 4L2 2
# a ÿ c2 a 2b c2 b ÿ c2 2a b c2 for d a b c, a c2 4L2 2 b c2 4L2 2 V 0 aa ÿ
1 p4 a2 , 4 L
A5
A6
where a, b, c and d are all dierent.
References Arnold, V.I., 1965. Small denominators I: Mappings of the circumference onto itself. Am. Math. Soc. Transl. Series 2 46, 213±284. Doedel, E.J., Wang, X.J., Fairgrieve, T.F., 1995. AUTO94: Software for continuation and bifurcation problems in ordinary dierential equations. Tech. rept. CRPC-95-2. California Institute of Technology. Gervais, J.J., Abderrahmann, O., Pierre, R., 1997. Finite element analysis of the buckling and mode jumping of a rectangular plate. Dynamics and Stability of Systems 12, 161±185. Golubitsky, M., Schaeer, D.G., 1984. Singularities and groups in bifurcation theory. In: Applied Mathematical Sciences, vol. 51. Springer-Verlag, New York. Hunt, G.W., Everall, P.R., 1999. Arnold tongues and mode jumping in the supercritical post-buckling of an archetypal elastic structure. Proc. R. Soc. Lond. A. 455, 125±140. Lord, G.L., Champneys, A.R., Hunt, G.W., 1997. Computation of localized post buckling in long axially-compressed cylindrical shells. In: Champneys, A.R., Hunt, G.W., Thompson, J. M.T. (Eds.), Localization and Solitary Waves in Solid Mechanics. Phil. Trans. R. Soc. Lond. 355, 2137±2150 (special issue). Maaskant, R., Roorda, J., 1992. Mode jumping in biaxially compressed plates. Int. J. Solids Structures 29 (10), 1209±1219. Nakamura, T., Uetani, K., 1979. The secondary buckling and post-secondary buckling behaviours of rectangular plates. Int. J. Mech. Sci. 21, 265±286. Schaeer, D.G., Golubitsky, M., 1979. Boundary conditions and mode jumping in the buckling of rectangular plates. Commun. Math. Phys. 69, 209±236. Sharman, P.W., Humpherson, J.G., 1968. An experimental and theoretical investigation of simply-supported thin plates subjected to lateral load and uniaxial compression. Aero. J. Roy. Aero. Soc. 72, 431±436. Supple, W.J., 1967. Coupled branching con®gurations in the elastic buckling of symmetric structural systems. Int. J. Mech. Sci. 9, 97±112.
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Supple, W.J., 1970. Changes of waveform of plates in the post-buckling range. Int. J. Solids Structures 6, 1243±1258. Thompson, J.M.T., Hunt, G.W., 1973. A general theory of elastic stability. Wiley, London. Thompson, J.M.T., Stewart, H.B., 1986. Nonlinear dynamics and chaos: geometrical methods for engineers and scientists. Wiley, Chichester. Uemura, M., Byon, O., 1977. Secondary buckling of a ¯at plate under uniaxial compression. Part 1: Theoretical analysis of simply supported ¯at plate. Int. J. Non-lin. Mech. 12 (6), 355±370. Uemura, M., Byon, O., 1978. Secondary buckling of a ¯at plate under uniaxial compression. Part 2: Analysis of clamped plate by FEM and comparison with experiments. Int. J. Non-lin. Mech. 13 (1), 1±12.
Arnold tongue predictions of secondary buckling in thin elastic plates P.R. Everall*, G.W. Hunt Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK Received 24 August 1998
Abstract The stability of post-buckled states for simply-supported ¯at elastic plates under compression is investigated for a range of in-plane boundary conditions. The von KaÂrmaÂn plate equations are reduced to a series of ODEs which are solved numerically under parametric variation of both load and length. Results are checked against full numerical solutions of the PDEs, and comparison with a modal analysis highlights the dominant passive contaminations. The nondimensional amplitude at secondary bifurcation, for any combination of modes and all plate lengths, is presented in a concise form using the parameter space of Arnold tongues. This demonstrates that compound bifurcation represents a worst case for post-buckling reserve, and that long plates have inherently more such reserve than short plates. It is also shown that stiening the boundaries against inplane movement is destabilizing, in that it induces mode jumping at secondary bifurcation to occur at an earlier stage in the post-buckling regime. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: B. Structures; C. Energy methods; Numerical algorithms; Mode jumping
1. Introduction Thin elastic plates have many applications which exploit their stable postbuckling response in compression. The aim of this paper is to investigate the limit of this stability for simply-supported ¯at plates subject to various in-plane boundary conditions. The post-buckling behaviour of such plates is sometimes * Corresponding author. 0022-5096/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 5 0 9 6 ( 9 9 ) 0 0 0 0 8 - 3
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explored using modal approximations (see, for example, Supple (1970), Uemura and Byon (1977) and Maaskant and Roorda (1992)). This has been successfully used to predict the type of coupled solution path (Supple, 1967), and the possibility of mode jumping, for plates with various boundary conditions. Results have been compared with experiments for one case (Sharman and Humpherson, 1968), although in general the limitations of the method have not been fully explored. Schaeer and Golubitsky (1979) use an analytical method to demonstrate mode jumping in straight-edged plates clamped in bending. Alternatively, Gervais et al. (1997) have shown the possibility of mode jumping in partially and fully clamped plates using a ®nite element method, but this turns out to be a time consuming process over a large range of aspect ratios and for dierent boundary conditions. Finite element and experimental results for a clamped plate have also been compared by Uemura and Byon (1978). This paper compares modal and numerical methods with a third approach, the reduction of the coupled partial dierential equations (PDEs) by the Galerkin procedure to a series of ordinary dierential equations (ODEs), which are solved using the boundary value solver AUTO (Doedel et al., 1995). This is successful because the mode interaction which leads to the loss of stability maintains a transverse pro®le that is similar at all positions along the length. Results are compared with full numerical solutions. Limitations of Rayleigh±Ritz (modal) analysis are also explored by comparing predictions of the secondary bifurcation points with those given by AUTO. The paper presents its ®ndings in the form of Arnold tongue plots (Arnold, 1965), borrowed from nonlinear dynamical systems theory (Thompson and Stewart, 1986). Previously presented for the ODE problem of a strut supported by a stabilizing elastic medium (Hunt and Everall, 1999), these summarize the coupled behaviour in a succinct manner.
2. Theory The post-buckling of a thin plate is described by the von KaÂrmaÂn largede¯ection equations, which in nondimensional form can be written r4 w wxx fyy ÿ p2 Px ÿ 2wxy fxy wyy fxx ÿ p2 Py ,
1
r4 f w2xy ÿ wxx wyy ,
2
where subscripts denote partial dierentiation with respect to the variable. Px and Py are the average applied loads per unit of length in the x and y directions, respectively, w is the vertical displacement and f is a stress function describing the change from the uniformly compressed state, de®ned by fyy
txy sy sx , fxy ÿ , fxx , E E E
3
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where stresses sx etc. again represent the change from uniform compression. The ®rst is an equilibrium and the second a compatibility equation. 2.1. Boundary conditions A simply-supported plate of unit width and length L is considered with three dierent sets of in-plane boundary conditions. For all three cases, shown in Fig. 1, there are no out-of-plane (z ) de¯ections or bending moments at the plate edges. A plate with no applied or reactive stresses at the boundaries, apart from the load Px, is shown in Fig. 1(a). All edges are free to move in the plane of the plate. In this case no secondary bifurcations occur on the natural loading path (Supple, 1970), mode jumping is absent and hence further analysis is not pursued. However, mode jumping has been observed when the edges are held straight; Fig. 1(b) shows that to achieve this, although Py=0, the stresses at the plate boundaries are no longer uniform. The variation from the uniformly compressed state is described by f, which has the following constraints: 1 fyy dy fxy 0 at x 0,L, 0
L 0
fxx dx fxy 0 at
y 0,1:
4
Fig. 1. Simply supported plate with: (a) edges free to move; (b) edges held straight; (c) long edges clamped in-plane. Broken lines show unloaded plate. Solid lines show post-buckled shape.
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For the plate that is clamped in-plane (Fig. 1(c)), such that the long boundaries are restrained from moving in the transverse direction but remain free to move longitudinally, the average transverse applied stress is non-zero. Before buckling it is clear that Py=nPx (where n is Poisson's ratio); however this no longer holds in the post-buckling regime, where 1 1 2 2 fxx ÿ nfyy ÿ wy dy: 5 p Py ÿ nPx 2 0 This is an integral condition across the width of the plate, and it is used directly in all the following numerical solutions to determine Py. Again, the boundary conditions for f are de®ned by (4). In the remainder of this paper we investigate the behaviour of plates subjected to the boundary conditions shown in Fig. 1(b) and (c). These will be termed `straight-edged' and `clamped in-plane' boundaries, respectively. In addition we will vary the single loading parameter, P 0Px. 2.2. Critical loads The critical loads can be found by linearizing (1) and (5), to give r4 w ÿwxx p2 P ÿ wyy np2 P
6
for the clamped in-plane boundary conditions of Fig. 1(c). Substituting w=A Lpx sin py, leads to the characteristic equation, L4 L2 P ÿ 2 1 ÿ nP 0: When 4 ÿ 4n < P < 1/n, L2 is real, s P ÿ 2 P ÿ 22 2 nP ÿ 1, L2 ÿ 2 4 leading to two imaginary eigenvalues de®ned by v s u u tP ÿ 2 P ÿ 22 ÿ nP ÿ 1, l1 2 4 v s u u tP ÿ 2 P ÿ 22 l2 nP ÿ 1: 4 2
7
8
9
(The two negative eigenvalues can be dropped because of the reversibility in x.) Over a ®nite length L however, to satisfy the simply-supported boundary conditions, the de¯ection must be an integer number of half-waves, n,
P.R. Everall, G.W. Hunt / J. Mech. Phys. Solids 47 (1999) 2187±2206
w A sin lpx sin py Qn sin
npx sin py, L
2191
10
and therefore l=n/L (where l is either l1 or l2). Substituting this value into (7) gives the critical loads for buckling into n half-waves at all plate lengths, Pn
n2 L2 2 : L2 n2 nL2
11
The values of l and Pn for the straight-edged boundary conditions of Fig. 1(b) can be found by substituting n=0 into (9) and (11), respectively. The variations in critical load for the ®rst four mode numbers are shown in Fig. 2(a) over a range of plate lengths. Critical loads for the clamped in-plane plate are lower than those for the straight-edged plate, with their minima occurring at relatively greater lengths. This is due to pre-buckled compression across the width, causing it to buckle at a load lower than when such compression is absent. At discrete plate lengths, the critical loads for dierent modes coincide at compound bifurcation points. Fig. 2(b) shows the variation of l1/l2 with load for the two cases. For the straight-edged plate, l1/l2 4 0 as P 4 1 and therefore compound bifurcation points can occur for all P > 4. However, if the edges are clamped in-plane, they are restricted to the range 4 ÿ 4n < P < 1/n; this is apparent from Fig. 2(a), where the critical loads for all mode numbers fall into a narrow band as the plate length increases. 2.3. A typical Arnold tongue Fig. 3(a) illustrates schematically the general form of the equilibrium solution paths for the following two modes (Supple, 1967),
Fig. 2. (a) Critical loads and (b) variation of l1/l2 with load. Solid line: straight-edged; broken line: clamped in-plane with n=0.3.
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apx bpx Qb sin w Qa sin L L
sin py,
12
where a < b, at a length just to the left of the corresponding compound bifurcation point of Fig. 2(a), where the curves of Pa and Pb cross. Bifurcation from the ¯at state occurs at the critical load Pa, whereupon the system buckles into a half-waves and follows the stable uncoupled path (Qb=0). The uncoupled path with b half-waves (Qa=0) intersects the fundamental path at Pb>Pa, and is therefore initially unstable with respect to Qa. Sa and Sb, respectively mark the positions where a coupled solution, for which Qa and Qb are both non-zero, intersects the lower and upper uncoupled paths. The coupled path projects onto the base plane as a closed loop as shown. The lower uncoupled path thus loses stability at Sa, where under dead loading conditions a dynamic jump takes place to the upper uncoupled path (Qa=0), which has stabilized at Sb. This is a mode jump from a to b half-waves. The position at which secondary buckling occurs for all plate lengths has been plotted as a function of load by Nakamura and Uetani (1979). This indicates the lowest secondary buckling load for any given plate length, but not the buckle pattern after mode jumping. In contrast, an Arnold tongues plot (Arnold, 1965) describes the secondary bifurcation positions for any two modes, at all plate lengths (Hunt and Everall, 1999). Arnold tongues were ®rst used to describe the behaviour of a second-order nonlinear forced oscillator (Thompson and Stewart, 1986). In a typical such system two frequencies are of interestÐa natural frequency and that of the forcing function. The tongues describe cusp-shaped loci which, in the space of the ratio of these frequencies and the amplitude of forcing, mark the limit of mode locking. For the plate, the ratio of the two natural `frequencies' is plotted against nondimensional amplitude (Fig. 3(b)), with the cusps marking the amplitudes of Sa and Sb for all plate lengths, and the ratio of
Fig. 3. (a) Equilibrium paths. Stable paths are shown as solid lines and unstable paths as broken lines. (b) Tongue for straight-edged plate, showing interaction between 2 and 3 half-waves.
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the interacting mode numbers being given by the point where the cusp meets the l1/l2 axis. For example, the tongue depicting the interaction between a = 2 and b = 3 for the straight-edged plate is plotted in Fig. 3(b). The values of both l1 and l2 for each of the two modes of interest are ®xed by its critical load, Pa or Pb (11). While one of these l values must be linked directly to an integer (a or b ), the other in general will not; typically therefore l1/l2 is irrational. Only when the length is such that two critical loads coincide will both l values be linked to integers, and l1/l2 be rational. The compound bifurcation point where Pa=Pb is thus marked by a rational position on the l1/l2 axis, in this case at l1/l2=2/3. Note also that the minimum position of each curve of Fig. 2(a) has l1=l2=1 and thus appears on the Arnold tongue plot at the right-hand margin. As we move to the left of the compound bifurcation point in Fig. 2(a), in this example at the intersection of the curves n = a=2 and n = b=3, Pb increases and hence the l1/l2 related to it decreases (Fig. 2(b)), while at the same time Sb emerges from the compound bifurcation point and moves up the uncoupled path. This is traced by the left-hand arm of the Arnold tongue, shown as a solid line in Fig. 3(b). The vertical axis gives the nondimensional amplitude Qb at the secondary bifurcation Sb. Of more practical interest is the position of Sa, represented by the right-hand arm of the cusp. Again to the left of the relevant intersection of Fig. 2(a), l1/l2 increases as Pa reduces, and the position of Sa is given by the solid line to the right of l1/l2=2/3. However, unlike Sb, Sa fails to pull in to the compound bifurcation point as Pa and Pb converge. Instead it remains remote from the fundamental path, and vanishes only when it collides with a further secondary bifurcation on the same uncoupled path, which is found to emerge from the compound bifurcation point in the region to the right of the n = 2 and n = 3 intersection of Fig. 2(a). (This complex mechanism is examined in detail in Hunt and Everall (1999).) These two secondary bifurcations are represented in Fig. 3(b) by the broken line in the right-hand arm of the tongue which initially veers to the left; as in this region Pa>Pb, they cannot be reached in a normal loading sequence. There is thus a certain threshold value of amplitude of Sa below which mode jumping is absent, which is picked out at l1/l2=a/b = 2/3, where the broken line changes to solid. Hence mode jumping only occurs at positions to the left of the relevant intersection of Fig. 2(a), and always involves a reduction in the wavelength of the buckle pattern, from a to b>a half-waves. p For the p straight-edged plate when a = 2 and b = 3, Pa=Pb occurs at L 6, so for L< 6 2:4 say, the coupled behaviour is similar to that shown in Fig. 3(a). The values of l1/l2 at Pa and Pb are now 0.694 and 0.64, respectively, shown by the vertical dotted lines in Fig. 3(b). The nondimensional amplitudes of the secondary bifurcation points are given by the intersection of these lines with the two arms of the cusp, in this example Qa 1 4.5 and Qb 1 0.4. No practicallyrelevant secondary bifurcation positions with Qa < 4.09 exist on the lower uncoupled path. In dimensional terms, where
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p 12 1 ÿ n2 qa , Qa t
13
this translates into an actual amplitude of qa 1 1.24t, where t is the plate thickness.
3. Analytical approaches 3.1. Numerical solution of the PDEs There are no simple exact solutions of the PDEs (1) and (2), so a classical ®nite dierence method is used to give a series of nonlinear algebraic equations. These equations are solved for w and f using an iterative method, the current solution values being substituted back into the algebraic equations and used to give an improved solution. This procedure is repeated until the change is less than a desired tolerance. Although this technique is simple to implement, it takes a long time to converge and does not locate secondary bifurcation points. A more sophisticated ®nite element program has been written (Gervais et al., 1997) which tracks the post-buckled paths and identi®es the secondary bifurcation points. However, this method remains time consuming, especially for large aspect ratios where many grid points are required to maintain accuracy. 3.2. Reduction to ODEs AUTO (Doedel et al., 1995) is a boundary value solver which is able to track solution paths and log secondary bifurcation points for ODEs, with much quicker solution times than for the PDEs. By assuming that the form of the de¯ection across the plate ( y ) remains the same at all positions along the length (see also Lord et al. (1997)), we can reduce the PDEs to a series of ODEs in x. The following functions w x,y w1 x sin py, f x,y f1 x f2 x cos 2py,
14
satisfy the boundary conditions on the long edges de®ned in (4) and (5). These are substituted into (1) and (2), and the Galerkin procedure is then used to produce a series of ODEs, which with the appropriate boundary conditions, are solved in AUTO. Unlike a Rayleigh±Ritz analysis which uses a limited number of modes, this method allows the interaction between all periodic forms along the length to be included. It has been shown (Nakamura and Uetani, 1979) that the eect of the ®rst transverse harmonic can be signi®cant at large amplitudes. This is included by using two displacement functions,
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w x,y w1 x sin py w2 x sin 3py, f x,y f1 x f2 x cos 2py f3 x cos 4py f4 x cos 6py:
15
However, as this doubles the number of ODEs that need to be solved, solution times will increase. The solutions of (14) and (15) are validated with the full numerical solution of the PDEs described above at discrete loads and aspect ratios as described in Section 4.1. 3.3. Rayleigh±Ritz The total potential energy, in nondimensional form, for an elastic plate of unit width and length L can be written (Lord et al., 1997), V
1 2
1 L
1 2
0 0
r2 w2 ÿ 2 1 ÿ n wxx wyy ÿ w2xy dxdy
1 L 0 0
1 ÿ p2 P 2
r2 f2 ÿ 2 1 n fxx fyy ÿ f2xy dx dy
1 L 0 0
16
w2x dx dy:
Sinusoidal functions along the length of the plate and the transverse functions given in (14) satisfy the boundary conditions (4) and (5). The stress function f is found by substituting these functions for w into (2) and integrating directly; compatibility is thus automatically satis®ed and a Lagrange multiplier is not required. These expressions for w and f are substituted into (16), leading to the energy function, V
1 XXXX 1X 0 Vabcd Qa Qb Qc Qd V aa P ÿ P a Q2a , 24 a b c d 2 a
17
where a, b, etc. are positive integers. Like the strut on a hardening foundation (Hunt and Everall, 1999), the energy function carries a fundamental symmetry in that it contains only even powers of w. Zero and non-zero Vabcd terms thus occur for the same combinations of a, b, c and d as for the strut, although the non-zero coecients are more complex (see Appendix A). Equilibrium solutions are given where all the ®rst variations of (17), @V , @Qa
@V , @Qb
18
etc., are simultaneously zero. Hence a ®rst order approximation of the load for the straight-edged plate is given by
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P Pa
a4 L4 Q2a : 16a2 L2
19
This modal approach can be used to predict the positions of the secondary bifurcations (Sa and Sb ) using a ®nite number of modes. The simplest approximation uses two modes (12), for which Supple (1967) shows that the amplitude at Sa is given by Qa
3V 0
6V 0 aa V 0 bb P b ÿ P a : 0 aa Vaabb ÿ V bb Vaaaa
20
However, previous work (Hunt and Everall, 1999) has shown that this is only accurate for small amplitudes. The agreement at large amplitudes can be improved by including other modes in the approximation; these are denoted by Greek letters and called passive modes because, in contrast to the `active' modes a and b, they have minimal eect close to the primary bifurcation points. When one passive mode is included, say a=2a ÿ b, the secondary bifurcation occurs where the determinant (Thompson and Hunt, 1973), 2 @ V=@ Q2 b @ 2 V=@ Qb @Qa
@ 2 V=@ Qb @ Qa , @ 2 V=@ Q2a
21
evaluated on the lower uncoupled path, vanishes. This analysis is essentially equivalent to a general Lyapunov±Schmidt reduction (Golubitsky and Schaeer, 1984). The approximation for the uncoupled path in Qa can be improved by including either the longitudinal, apx 3apx Qb sin sin py, w Qa sin L L
22
or both the longitudinal and transverse harmonics, apx 3apx Qb sin sin py w Qa sin L L apx 3apx Qd sin sin 3py, Qg sin L L
23
although (17) and (21) both increase in complexity. Again, the solutions can be validated against the numerical solution, and the secondary bifurcation positions given by this analysis compared with those from AUTO. Therefore, unlike other work (Supple, 1970; Uemura and Byon, 1977; Maaskant and Roorda, 1992), the dominant harmonics and passive modes can be identi®ed and the limitations of this approach assessed.
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4. Results 4.1. Validation 4.1.1. Uncoupled paths The solutions given by the three methods described above are compared by calculating the total energy (16) of a square plate (L = 1) at dierent loads. Fig. 4(a) shows the percentage dierence between the energy of the numerical solution and values given by the AUTO and Rayleigh±Ritz methods, for the straight-edged plate. The solution of (14) by AUTO has an error of approximately 3% at 3Pn, but with inclusion of the transverse harmonic (15) this reduces to 0.1%, giving excellent agreement with the numerical solution. Solutions of a single mode Rayleigh±Ritz analysis are poor, but addition of the ®rst harmonic (22) gives solutions very similar to the same approximation in AUTO. Solutions of the four mode approximation (23) are hard to distinguish from both the numerical solution and those arising from the same transverse variation in AUTO. This implies that uncoupled paths can be accurately modelled by these four modes. Fig. 4(b) shows similar trends for a plate with the clamped in-plane boundary conditions. 4.1.2. Secondary bifurcations Like the numerical solution of the PDEs, locating the secondary bifurcations using AUTO for long plates is time consuming. Therefore AUTO was also used to validate the predictions given by the Rayleigh±Ritz approximation, which gives solutions more quickly. The dierences between AUTO and three modal approximations are summarized by the tongue describing the interaction between two and three half-waves, shown in Fig. 5 for the two sets of boundary conditions. The simplest approximation plotted includes a, b and one passive
Fig. 4. Percentage errors in energy (16) between numerical solution and: Rayleigh±Ritz solution of (17) with (i) 1, (ii) 2 and (iii) 4 modes (broken line); AUTO solution of (iv) (14) and (v) (15) (solid line). (a) Straight-edged; (b) clamped in-plane.
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contamination, a=2a ÿ b; Fig. 5(b) shows this only gives good agreement for Qn < 1. This is expected, as the single mode approximation for the uncoupled path (curve (i) of Fig. 4(b)) is poor. When the eects of the ®rst harmonic (22) and two other passive modes (2a+b and 4a ÿ b ) are introduced, acceptable agreement extends to Qn 1 3 (Fig. 5(a)). For the clamped in-plane boundary conditions, secondary buckling occurs at relatively low loads where the error using this approximation is less than 1% (curve (ii) of Fig. 4(b)); Fig. 5(b) shows that this is indistinguishable from the numerical results. This approximation also gives accurate predictions for Sb, as the amplitudes of the secondary bifurcations on the upper uncoupled path are also relatively small. Fig. 5(a) shows that inclusion of the transverse harmonic (23) in conjunction with the same passive modes (3a, 2a ÿ b, 2a+b and 4a ÿ b ), gives accurate predictions when Qn < 6. However, this gives a very lengthy energy function (17) and a 64 element stability matrix (21); both are dicult to manipulate and time consuming to solve. Therefore, for the straight-edged plate, a modal approximation of sucient accuracy is not signi®cantly quicker than AUTO. It also highlights the increase in the number of passive modes as Sa moves away from the primary bifurcation point. 4.2. Arnold tongues The Arnold tongues for plates with straight-edged and clamped in-plane boundary conditions are plotted in Fig. 6(a) and (b), respectively, using the solutions of (15) given by AUTO. Most secondary bifurcations occur on uncoupled paths which are already unstable with respect to another mode and therefore have no practical signi®cance. For clarity, only the ®rst four tongues which mark secondary bifurcations associated with the initial loss of stability are shown. As described in Section 2.3, the tongues always meet the l1/l2 axis at rational positions, giving the ratio between the number of half-waves before and after mode jumping, a/b. For the clamped in-plane plate (Fig. 6(b)), the fact that
Fig. 5. Comparison between numerically (solid line) and modally (broken line) produced tongues with (i) 3, (ii) 6, (iii) 12 modes. (a) Straight-edged; (b) clamped in-plane.
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Fig. 6. Arnold tongues produced from AUTO results (a) straight-edged; (b) clamped in-plane.
the tongues do not cross suggests that the case of practical relevance is always b=a + 1. However, for the straight-edged plate (Fig. 6(a)), some of the righthand arms do cross, indicating that mode jumping may take place with, say, the mode b=a + 2. All the illustrated tongues curve back on themselves, as seen in Fig. 3, and so the smallest amplitude of secondary buckling is given when l1/l2=a/b (and hence Pa=Pb ) for each combination of modes; fully-coincident buckling thus reduces the safety margin in the post-buckling range, as might be expected. The locus of these points, shown as a dotted line, is thus a safety envelope, inside which no secondary buckling can occur. Tongues with low values of l1/l2, for example 1/4, are outside the safety envelope, with the right hand arm moving initially to the right, so mode jumping could occur at vanishingly-small amplitudes just above Pa (Hunt and Everall, 1999). However, these interactions occur on unstable uncoupled paths, and are not practically relevant. What is perhaps more surprising is that for each of the plots of Fig. 6, the safety envelope increases in the Qn sense as l1/l2 4 1. This implies that, as the plate gets longer and the interacting wavenumbers get relatively closer together, secondary bifurcation is increasingly delayed. This perhaps seems counterintuitive, as it might be thought that the interactions present in shorter plates, where the boundary conditions place considerable restriction on the choice of wave, would always be available to the less restricted longer plate. That this is not the case is attributable to sympathetic mode locking. Consider for example the interaction between a = 4 and b = 5, for the plate that is clamped in-plane. The most severe interaction takes place when l1/l2=4/5 on the Arnold tongue picture of Fig. 6(b). However it is not the case that this particular interaction would occur in a plate of exactly double the length. The critical load plot of Fig. 2(a) reveals that at this length the plate would buckle into a = 9 half-waves, and subsequently interact with b = 10. The worst-case interaction for this latter problem is now at l1/l2=9/10 on the Arnold tongue
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picture, with the safety envelope showing that secondary bifurcation is inevitably further delayed. Note that the Arnold tongues plot indicates the positions of secondary bifurcations, whereas stability may also be lost at a limit point, either before or after secondary buckling. The complete picture of the coupled paths, which may be distorted greatly by passive eects, is now explored using AUTO. 4.3. Passive mode eects 4.3.1. Limit point instability Even at relatively small amplitudes the eect of the passive modes can be signi®cant; when the primary mode is a single half-wave for example, the amplitude of the ®rst passive harmonic (b=3a = 3) also becomes large when Qa is large. Fig. 7 shows the contaminating mechanism by which stability is lost to this harmonic, before the secondary bifurcation marking interaction with a two halfwave solution is encountered. The contamination is due to Vaaab, which, being negative (see Appendix A), is destabilizing in nature, whereas Vaabb is positive and stabilizing. Thus stability is lost at a limit point which is not picked out by the Arnold tongues plot, and accounts for the absence of a tongue at l1/l2=1/2 for both sets of boundary conditions. 4.3.2. Stable secondary bifurcation For the clamped in-plane plate, secondary buckling from a = 2 to b = 3 occurs before the uncoupled path is destabilized by the passive harmonic b=3a = 6. However, Fig. 8(a) shows that at Sa the coupled path is initially stable. Mode jumping is then delayed until a limit point where P > PSa; again, in this instance, the tongue does not show the amplitude where stability is lost. For slightly shorter lengths (Fig. 8(b)), the isolated solution path of Fig. 8(a) merges with the coupled path. This gives a very complex coupled path, with the amplitudes of some passive modes being greater than Qa and Qb, highlighting the increasing eect of passive
Fig. 7. Eect of ®rst harmonic at Pa=Pb for the clamped in-plane plate. Stable paths are shown as solid lines and unstable paths as broken lines.
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modes with displacement away from both primary and secondary bifurcation points. Stability is now lost at a higher load than might be expected from Fig. 8(a), with the path restabilizing before a jump to the 3 half-wave solution occurs. 4.3.3. Unstable secondary bifurcation As L increases, the coupled path moves from stable to unstable. Fig. 9(a) shows that mode jumping occurs at the secondary bifurcation point when a = 4. For slightly shorter lengths (Fig. 9(b)) the coupled path again merges with the isolated solution path, and although it restabilizes the load reached is less than PSa, implying that under dead loading conditions the buckle pattern will change instantaneously. Therefore the tongues mark the limit of stability when a r4. 4.3.4. Crossing of tongues The amplitudes of Sa for the straight-edged plate are much larger than for clamped in-plane boundary conditions, and hence the eect of the passive modes will be greater and the coupled behaviour more complex. The tongue of l1/l2= 3/5 crosses that of l1/l2=3/4, showing that when a = 3 interaction can occur with the mode b=a + 2=5; this coupled path contains only odd modes which have stronger coupling (V3355>V3344 for example). When b=a + 1, the coupled path is symmetrical with respect to both Qa and Qb. However, when a and b are both odd this double-symmetry is broken; ¯ipping one mode through while keeping the other unchanged generates a dierent shape. This is re¯ected in the plot of Fig. 10, in which the coupled path does not project onto the base plane as a closed loop. 5. Conclusions The paper explores the eect of the in-plane boundary conditions on the postbuckled response of a simply-supported ¯at plate. As the boundaries are
Fig. 8. Jump from 2 to 3 waves for the clamped in-plane plate, (a) L = 3.80, (b) L = 3.76. Stable paths are shown as solid lines and unstable paths as broken lines.
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Fig. 9. Jump from 4 to 5 waves for the clamped in-plane plate, (a) L = 6.70; (b) L = 6.60. Stable paths are shown as solid lines and unstable paths as broken lines.
increasingly constrained against in-plane movement, the amplitude, and hence the load, at which secondary bifurcation occurs reduces: the unrestrained plate of Fig. 1(a) is unconditionally stable; for most lengths the straight-edged plate loses stability at P > 2Pn; the plate with its edges clamped in-plane exhibits little by comparison in the way of post-buckled strength. The most useful results are obtained by reducing the von KaÂrmaÂn equations to a set of ODEs which are solved numerically using AUTO, hence including the destabilizing eects of all longitudinal harmonics and passive modes. Validation with the full numerical solution of the PDEs highlights the eect of transverse harmonics, especially at large amplitudes. Comparisons with Rayleigh±Ritz results show that passive modes can signi®cantly aect the amplitude of secondary buckling, so the coupled behaviour can be complex. The results are presented using the parameter space of Arnold tongues. This plot shows the amplitude at which secondary buckling occurs for all plate lengths,
Fig. 10. Jump from 3 to 5 waves for the straight-edged plate. Stable paths are shown as solid lines and unstable paths as broken lines.
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with the point where the tongue meets the l1/l2 axis giving the ratio between the wavelengths before and after mode jumping. The safety envelope marks the smallest amplitude at which secondary buckling can occurÐit is noted with interest that this amplitude increases as L 4 1. However, the tongue picture does not indicate the behaviour or stability of the coupled path; in general for short plates the coupled path is stable, with mode jumping only occurring at the secondary bifurcation point for larger aspect ratios. In most practical applications ¯at plates are not simply-supported, and therefore it might be interesting to extend this work to encompass other than simply-supported boundaries. The post-buckled wavelength can now modulate with length, and hence it may not be possible to present the results directly in the form of Arnold tongues.
Acknowledgements The authors are grateful to Gabriel Lord for supplying a working version of his Galerkin reduction. This work has been ®nanced by an Engineering and Physical Sciences Research Council (EPSRC) research studentship.
Appendix A. Energy coecients An algebraic manipulation package was used to calculate the energy coecients in (16). The non-zero values for the straight-edged plate are, 3 4 a4 A1 Vaaaa p L 1 4 , L 32
Vaabb
" # 1 4 a2 b2 a ÿ b4 a b4 p L 4 4 , L 32 a b2 4L2 2 a ÿ b2 4L2 2
Vaaab ÿ
3 4 p L 32
for
b 3a,
A2
A3
Vaabc
" # 1 4 a b2 3a ÿ b2 p L 3 32 a ÿ b2 4L2 2
Vaabc
" # 1 4 a ÿ b2 3a b2 ÿ p L 3 for 32 a b2 4L2 2
for
c 2a ÿ b,
c 2a b,
A4
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Vabcd
" 1 4 a ÿ b2 a b 2c2 p L 6 64 a b2 4L2 2
# a ÿ c2 a 2b c2 b ÿ c2 2a b c2 for d ÿa ÿ b ÿ c, a c2 4L2 2 b c2 4L2 2
Vabcd
" 1 4 a ÿ b2 a b ÿ 2c2 ÿ p L 6 64 a b2 4L2 2
# a c2 a 2b ÿ c2 b c2 2a b ÿ c2 for d ÿa ÿ b c, a ÿ c2 4L2 2 b ÿ c2 4L2 2
Vabcd
" 1 4 a b2 a ÿ b 2c2 ÿ p L 6 64 a ÿ b2 4L2 2
# a ÿ c2 a ÿ 2b c2 b c2 2a ÿ b c2 for d ÿa b ÿ c, a c2 4L2 2 b ÿ c2 4L2 2
Vabcd
" 1 4 a b2 a ÿ b ÿ 2c2 p L 6 64 a ÿ b2 4L2 2
# a c2 a ÿ 2b ÿ c2 b ÿ c2 2a ÿ b ÿ c2 for d ÿa b c, a ÿ c2 4L2 2 b c2 4L2 2
Vabcd
" 1 4 a b2 a ÿ b ÿ 2c2 ÿ p L 6 64 a ÿ b2 4L2 2
# a c2 a ÿ 2b ÿ c2 b ÿ c2 2a ÿ b ÿ c2 for a ÿ c2 4L2 2 b c2 4L2 2
Vabcd
d a ÿ b ÿ c,
" 1 4 a b2 a ÿ b 2c2 p L 6 64 a ÿ b2 4L2 2
# a ÿ c2 a ÿ 2b c2 b c2 2a ÿ b c2 for d a ÿ b c, a c2 4L2 2 b ÿ c2 4L2 2
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"
Vabcd
1 4 a ÿ b2 a b ÿ 2c2 p L 6 64 a b2 4L2 2
# a c2 a 2b ÿ c2 b c2 2a b ÿ c2 for d a b ÿ c, a ÿ c2 4L2 2 b ÿ c2 4L2 2
Vabcd
" 1 4 a ÿ b2 a b 2c2 ÿ p L 6 64 a b2 4L2 2
# a ÿ c2 a 2b c2 b ÿ c2 2a b c2 for d a b c, a c2 4L2 2 b c2 4L2 2 V 0 aa ÿ
1 p4 a2 , 4 L
A5
A6
where a, b, c and d are all dierent.
References Arnold, V.I., 1965. Small denominators I: Mappings of the circumference onto itself. Am. Math. Soc. Transl. Series 2 46, 213±284. Doedel, E.J., Wang, X.J., Fairgrieve, T.F., 1995. AUTO94: Software for continuation and bifurcation problems in ordinary dierential equations. Tech. rept. CRPC-95-2. California Institute of Technology. Gervais, J.J., Abderrahmann, O., Pierre, R., 1997. Finite element analysis of the buckling and mode jumping of a rectangular plate. Dynamics and Stability of Systems 12, 161±185. Golubitsky, M., Schaeer, D.G., 1984. Singularities and groups in bifurcation theory. In: Applied Mathematical Sciences, vol. 51. Springer-Verlag, New York. Hunt, G.W., Everall, P.R., 1999. Arnold tongues and mode jumping in the supercritical post-buckling of an archetypal elastic structure. Proc. R. Soc. Lond. A. 455, 125±140. Lord, G.L., Champneys, A.R., Hunt, G.W., 1997. Computation of localized post buckling in long axially-compressed cylindrical shells. In: Champneys, A.R., Hunt, G.W., Thompson, J. M.T. (Eds.), Localization and Solitary Waves in Solid Mechanics. Phil. Trans. R. Soc. Lond. 355, 2137±2150 (special issue). Maaskant, R., Roorda, J., 1992. Mode jumping in biaxially compressed plates. Int. J. Solids Structures 29 (10), 1209±1219. Nakamura, T., Uetani, K., 1979. The secondary buckling and post-secondary buckling behaviours of rectangular plates. Int. J. Mech. Sci. 21, 265±286. Schaeer, D.G., Golubitsky, M., 1979. Boundary conditions and mode jumping in the buckling of rectangular plates. Commun. Math. Phys. 69, 209±236. Sharman, P.W., Humpherson, J.G., 1968. An experimental and theoretical investigation of simply-supported thin plates subjected to lateral load and uniaxial compression. Aero. J. Roy. Aero. Soc. 72, 431±436. Supple, W.J., 1967. Coupled branching con®gurations in the elastic buckling of symmetric structural systems. Int. J. Mech. Sci. 9, 97±112.
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Supple, W.J., 1970. Changes of waveform of plates in the post-buckling range. Int. J. Solids Structures 6, 1243±1258. Thompson, J.M.T., Hunt, G.W., 1973. A general theory of elastic stability. Wiley, London. Thompson, J.M.T., Stewart, H.B., 1986. Nonlinear dynamics and chaos: geometrical methods for engineers and scientists. Wiley, Chichester. Uemura, M., Byon, O., 1977. Secondary buckling of a ¯at plate under uniaxial compression. Part 1: Theoretical analysis of simply supported ¯at plate. Int. J. Non-lin. Mech. 12 (6), 355±370. Uemura, M., Byon, O., 1978. Secondary buckling of a ¯at plate under uniaxial compression. Part 2: Analysis of clamped plate by FEM and comparison with experiments. Int. J. Non-lin. Mech. 13 (1), 1±12.