RASD 20th11 International Conference

1-3 July 2013Pisa 3

ON THE BEHAVIOR OF POST-BUCKLED PLATES

Lawrence N. Virgin1* and Ted Lyman2

1Department of Mechanical EngineeringDuke UniversityDurham, NC 27708, USAE-mail: l.virgin@duke.edu

2Post-Doctoral Research FellowLos Alamos National LabLos Alamos, NM, USAE-mail: tcl5@duke.edu

Keywords: vibration, post-buckling, plates, structures, continuation

ABSTRACT

When loaded beyond their initial critical loads, plates continue to exhibit positive stiffness,and it is useful to gain some insight into possible equilibrium configurations, natural fre-quencies and vibration mode shapes in the post-buckled regime. In this paper use is madeof continuation methods, whereby a solution to an underlying problem is tracked includinginto the heavily post-buckled regime. The stability of resulting equilibrium paths is automat-ically assessed since the method relies on the computation of local Jacobian matrices. Whenpaired with a Galerkin approximation, continuation methods are shown to be well suited tosolving non-linear buckling problems. In addition to providing a robust solution methodfor nonlinear equations, it is relatively easy to extend the continuation approach to extractnatural frequency and mode shape information. Using the continuation package AUTO, pri-mary and remote secondary equilibrium branches are identified, and followed as a functionof axial load. The effect of initial geometric imperfections are included together with somecomparisons with experimental data.

1. INTRODUCTION

Lightweight aerospace structures frequently have structural components that may operateinto the post-buckled range. A wide class of structure can be categorized as flat plates (seee.g., [1–7]). An especially interesting phenomenon is mode jumping, where the natural post-buckled path becomes unstable and the structure dynamically snaps to a remote equilibrium[8–10]. However, it is also possible that a remote equilibrium configuration that, despite beingstable, is only realized if the system is subject to a large perturbation. If inertia is considered,there is also an interplay between the change in stiffness of the structure and its dynamicmodal characteristics [11–13]. For post-buckled plates, linearized natural frequencies andmode shapes can be extracted for small perturbations about nonlinear static deflections. Thenumerical method of continuation is utilized in this paper [14, 15].

2. BACKGROUND

The von Karman equations are used as the basis for plate analysis in this paper. Since theyare one of the standard approaches in the analysis of plates, and because they cannot beintroduced briefly, we simply state that the governing equations are set-up such that they canbe manipulated using a Galerkin technique in terms of a Fourier expansion associated withassumed deflection (that satisfies certain boundary conditions). More details can be found in[6]. The resulting algebraic equations are then solved using a continuation approach. Again,in the interests of space this standard technique is not covered in detail here, and the interestedreader is referred to [15]. Suffice it to say that this approach basically involves path-followingnonlinear solutions x satisfying

x = f(x, λ) (1)

in which λ is the control parameter. In the current context x relates to the coefficients de-scribing the deflected shape (w = w(x)) and λ is the axial load. This approach is related toNewton-Raphson with a corrector-predictor step in order to track a solution (wherever it maygo). The open-source package AUTO is used [14].

3. THE UNDERLYING FUNDAMENTAL BEHAVIOR

We shall focus on one of the simplest cases: a thin, uniaxially-loaded, simply-supportedsquare plate. A schematic picture is shown in Fig. 1(a): Under these conditions the axial

Figure 1: (a) A simply-supported square panel, (b) the classical super-critical pitchfork bifur-cation at buckling, (c) dependence of the natural frequency on the axial load.

load Px is gradually increased until a critical value is encountered (Pcr) where the flat con-figuration becomes unstable and the structure buckles (w/ne0) into one of two symmetricconfigurations. This is shown schematically in part (b), the normal form of which is the clas-sical super-critical pitchfork bifurcation. The potential energy, V , at a specific post-criticalloading level, is shown superimposed and we see the classic ’double-well’ shape with thetwo minima corresponding to the co-existing post-buckled equilibria. In part (c) is shownhow the natural frequency drops to zero at the critical point, but then starts to increase as thepost-buckled equilibrium path is followed. At this level of axial load the unstable trivial equi-librium has an imaginary natural frequency (associated with negative stiffness). If the systemis perfectly symmetric, then the linear natural frequencies within each potential energy wellare equal, at a given load level.

3.1 Effect of initial geometric imperfections

It is well-known that the behavior of axially-loaded structures in general, and buckling in par-ticular, tends to be heavily influenced by small geometric effects, especially those that tend tobreak symmetry [12]. Fig. 2(a) shows this behavior schematically. Here, there is a preferred

deflection under axial load as the system is biased to deflect in a particular direction, for theequilibrium paths w , −w at a given load. We also see that the (squared) natural frequencynow no longer drops to zero but rather veers to a small value in the vicinity of buckling be-fore increasing into the post-buckled range. The natural frequency of oscillations still occurabout either post-buckled equilibria, although they are slightly different. The complementaryremote equilibrium solution would only be picked up if the system were pulled to the side ofnegative deflections, and these would also be accompanied by unstable paths and imaginaryfrequencies but these are omitted from the plot. The label Pcr is used to indicate the critical

Figure 2: (a) The effect of an initial geometric imperfection, (b) the behavior of the funda-mental natural frequency through buckling.

load for the initially perfect case. There are, of course, higher ’critical’ buckling loads butsince the structure buckles at the lowest value, these are (apparently) not of much practicalvalue. However, as we shall see, they still have a role to play for highly buckled behaviorunder large perturbations.

4. RESULTS FOR THE FULL PLATE

Using three modes in each direction the full plate equations were solved. Fig. 3(a) shows theprimary post-buckled branch (the unstable equilibrium branch is shown as a dashed line), butonly positive deflections are plotted. Also shown is the case in which the plate has an initial

Figure 3: (a) The primary equilibrium paths for the perfect (black) and imperfect (red) plate,(b) the lowest four natural frequencies as a function of axial load.

curvature (in the positive deflection) corresponding to the plate thickness. The complemen-

tary paths and corresponding natural frequencies are not included in this plot. As expectedthe distinct nature of the instability has been lost and the plate deflection gradually increaseswith axial load.

Part (b) of this figure shows the lowest four natural frequencies. We see a gradual droppingof the lowest (fundamental) frequency to zero at buckling. The lowest frequency (now associ-ated with the non-trivial solution branch) then increases in to the post-buckled range. This isthe same form of behavior seen schematically in Fig. 2(b). The next three lowest frequenciesare also plotted. The corresponding natural frequencies about the imperfect equilibrium path(shown in red) are shown also. Given the relatively large value of the initial imperfection,these frequencies hardly reduce at all, i.e., there is considerable (and potentially useable)post-buckled stiffness in the plate.

Fig. 4 shows the equilibrium path originating from the second lowest critical point on thetrivial path. It is interesting to observe that this path later stabilizes, such that for an axial loadof more than twice the initial critical load the plate appears to have two stable equilibria (forpositive directions of the deflection). Part (b) shows the corresponding imperfect result, and

Figure 4: (a) The first two non-trivial equilibrium paths for the perfect plate, (b) the corre-sponding paths for the imperfect plate, (b) the lowest four natural frequencies as a functionof axial load for the new remote path, perfect (black), and imperfect (red).

although the new stable branch has been pushed to higher loads it still persists. In this figurethe ’deflection’ of the plate is couched in terms of the L2-norm of the Fourier coefficients toavoid nodal lines. Part (c) shows the first four natural frequencies about the new equilibriumbranch (only) for both perfect (black) and imperfect (red) geometries.

Suppose we have an axial load acting that is about four times as high as the initial criticalload for the geometrically perfect configuration. If the load had been monotonically increasedfrom zero (the typical case) then the deflection would follow the natural primary path withan L2-norm developing close to 12 (for either the perfect or imperfect cases, and indicatedby the right large data point in parts (a) and (b). However, while maintaining the level of thisaxial loading the plate can be pushed onto the other co-existing equilibrium configuration(with an L2-norm growing to about six). These are indicated by the left data points in parts(a) and (b)). Thus we see that although the second lowest bifurcation from the unstable trivialbranch, which initiates a path that is itself initially unstable, the subsequent re-stabilizationhas physical meaning if a sufficiently large disturbance is given to the system. That is, thereare at least two (and possibly four) co-existing equilibrium configurations for the plate atsufficiently large axial compression.

5. EXPERIMENTAL RESULTS

Some preliminary experimental studies were conducted. Fig. 5 shows some relevant details.The plate was made of polycarbonate (Young’s modulus of 2.4 GPa and density of 1.2 g/cm3)of dimensions 304.8 mm by 304.8 mm with a thickness of 1.55 mm. The boundary conditionswere designed to produce as close to pinned as possible. The vertical edges consisted of aseries of two columns of Delrin spheres (part (b)) and V-grooves along the top and bottom(the loaded edges). Axial loading was supplied by adding weights to the top edge. The out-

Figure 5: The experimental configuration, (a) the (polycarbonate) plate mounted in the testrig, (b) a detail of the simply-supported boundary conditions (on the vertical edges), (c) mea-surement locations.

of-plane deflection was measured using a Micro-Epsilon optoNCDT 1302 laser displacementsensor, located at the three points indicated in part (c).

Fig. 6 shows a summary of the experimental results. A Southwell plot technique [12] wasused to estimate an initial geometric deflection of amplitude 0.18h. Part (a) of the figuredisplays the primary equilibrium branch for the (necessarily) imperfect geometry. The platedeflection grows with the magnitude of the axial load analogous to the analytical resultsshown in Figs. 3 and 4. A typical deflected shape (at Px/Pcr = 4.35) is shown in the upperright inset. However, the other three equilibrium configurations were obtained by manuallypulling the plate across the (unstable) flat configuration such that the other configurationsare found. For example the ’inner’ branches were obtained by appropriately ousting andpulling on the plate in order to promote the second-mode shape that remains stable (once ithas been located). We see that there are coexisting (full sine wave corresponding to a modetwo) deflections on either side. These are the branches emanating from the already unstableflat configuration, but as mentioned earlier, would not have been obtained under a natural(monotonically increasing) loading path. Two sets of data points are shown for each pathsince data was acquired during unloading as well as loading.

Unfortunately the damping present in the experimental system did not allow any naturalfrequencies to be extracted from the experimental system in terms of free vibration undersignificant axial loading. The pinned boundary conditions tended to provide a degree of un-wanted rotational restraint with axial load. In fact, this is not unexpected in of the theoreticalmolding of damping effects [17], in addition to the practical issue of maintaining appropriateboundary conditions. However, some forced vibration testing was able to confirm some ofthe natural frequencies under conditions of light axial loading.

Figure 6: (a) The equilibrium paths of the plate including the ’remote’ path. The solid linesare AUTO, data points from experiment, (b) The deflected shapes provide snapshots alongthe vertical centerline for Px/Pcr = 4.35. Dashed curves were obtained from AUTO, Solidcurves with data points from experiment.

6. CONCLUSIONS

This paper reported on a continuation-based approach to obtaining all the equilibrium con-figurations of an imperfect, uniaxailly-loaded, simply-supported thin panel, including deflec-tions deep into the post-buckled regime. An interesting remote solution associated with are-stabilized branch emanating from a higher buckling mode was identified. This approachhas also been shown to work well for plates with other boundary conditions, cylindrical shells,and also in order to obtain natural frequencies and vibration modes. Some limited experimen-tal studies confirmed much of the analysis, although increased damping under axial loadingprevented extraction of usuable natural frequencies.

ACKNOWLEDGMENTS

TCL was supported by a NASA GSRP Fellowship.

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