Elsevier

Journal of Sound and Vibration

Volume 439, 20 January 2019, Pages 300-309
Journal of Sound and Vibration

Experiments on the thermal post-buckling of panels, including localized heating

https://doi.org/10.1016/j.jsv.2018.08.043Get rights and content

Abstract

The suppression of expansion in thin clamped panels subjected to elevated thermal loading often results in buckling. However, a number of possible post-buckled equilibrium configurations typically exist, and which shape ensues depends on a number of factors including the role of symmetry, boundary conditions, aspect ratio, and the effect of small geometric imperfections associated with the initial shape. It is possible to force the panel to go between different buckled shapes, given a sufficiently large perturbation. Sometimes the panel will spontaneously jump, or snap, when the temperature is gradually increased or decreased (mode jumping). The extent to which these features occur when the thermal loading is applied locally is also investigated. This paper describes some interesting nonlinear (essentially buckling) behavior in thermally-loaded panels from a primarily experimental perspective, with an additional focus on non-uniform heating. The full force of stereo 3D digital image correlation and forward-looking infrared cameras are exploited to provide a relatively complete picture of this behavior.

Introduction

It is not uncommon in problems of axially-loaded slender structures with nominal geometric and loading symmetry to observe an uncertain direction of post-buckling deflection [1]. Even the pin-ended Euler strut, the most classical example of buckling, presents a system in which the direction of the buckling depends on the presence of even very small geometric imperfections or perhaps a tiny axial load eccentricity, that has the tendency to break the symmetry [2,3]. Thus, a preferred direction is followed under the slow increase of loading through the buckling process. However, this does not mean to say that the complementary (alternative) equilibrium path - the path not naturally followed, does not exist: it simply requires a relatively large perturbation to deviate from the natural loading path [4]. That is, it is possible to physically push the strut from one equilibrium configuration to its opposite side. This is not dissimilar to the frequency sweep in dynamics (for example, through a nonlinear resonance), in which the response followed is typically associated with a local evolution, in which any remote response is not necessarily picked-up. This path dependency is dramatically different from a linear system with its unique equilibrium configuration and response.

When the structural system is a nominally flat panel or plate the same situation occurs, only now, further boundary conditions and the aspect ratio of the panel provide additional geometric variables. For relatively elongated plates, the aspect ratio is relatively large (in contrast to a square plate for example), and it is possible for the plate to exhibit a number of different modes, able to persist in their stable state [5]. Again, a large (judiciously chosen) perturbation can be applied to carry the system between these equilibria. In some circumstances the plate jumps between equilibria quite naturally under slowly changing loading, typically when a local instability is reached, (often a signal of hysteresis), and this type of instability behavior has been called mode jumping in the literature [[6], [7], [8], [9]]. Otherwise, under a nominally fixed axial load (temperature) any alternative, co-existing equilibrium shapes can be revealed with the application of an appropriate external force, and this is a dominant feature of the current study.

Thermal loading often results in compressive axial loading in a constrained slender structure, and many studies have been conducted into the buckling of such systems [10,11]. Some of this research has focused on analytic approaches [[12], [13], [14], [15]], and there has been much recent interest in the thermal loading of composite structures [[16], [17], [18], [19], [20]], the influence of boundary conditions [[20], [21], [22], [23]], circular plates [24], dynamics [8,9,23], and localized heating [25,26]. The current paper considers the thermal buckling of plates focusing on the following features:

  • a number of different materials (metals),

  • two sets of boundary conditions,

  • uniform and localized heating.

In addition, full-field data was acquired using the latest digital, non-invasive measurement techniques:
  • FLIR for temperature distribution,

  • DIC for displacement measurements.

The rest of this paper describes experimental results from tests in which the primary focus was to illustrate the existence of multiple equilibrium configurations in the buckled shape of thermally loaded plates. A key component of this investigation is the use of a local probe, enabling the panel to be laterally pushed between equilibria - which in turn depend on the level of thermal loading. The probe (a wooden rod) is used purely as a means to an end: its effect is removed immediately after each perturbation, such that all observed behavior is associated with equilibrium configurations, perhaps preceded by a little transient dynamic behavior.

A total of ten 254 × 94 mm panels (width-to-height ratio of 2.71:1) were examined in this work. The ten panels can be further classified by the five materials examined: aluminum, bronze, brass, mild steel and stainless steel as shown in Fig. 1(a). The thickness of the sheets of metal all varied by a small amount but the main results presented here (2024-T3 aluminum and C510 bronze) were both 0.813 mm thick (0.032 in). Two relatively stiff aluminum frames were machined to provide different boundary conditions for each material examined. Each panel was clamped into one of the two aluminum frames as schematically represented in Fig. 1(b). The first frame was machined to provide clamped boundary conditions on all four sides and the second frame was machined to provide clamped boundary conditions on side 1 and 2 with simply supported boundaries on sides 3 and 4 (V-grooves). Although the qualitative nature of the results to be reported later did not significantly change with repeating data acquisition after re-clamping, it should be noted that there was some sensitivity to the details of the clamping. Most axially-loaded structures tend to be sensitive to geometric imperfections and boundary conditions [3]. These effects were minimized by clamping the same number of bolts and to the same torque. However, placing the long edges of the panel into the V-grooves clearly involved some inaccuracies, and these issues would present challenges for theoretical modeling.

Due to the relative similarity of the materials (and the fact that not all the metals are amenable to the localized heating mechanism, to be described later) only a subset of the total results will be presented here, with a primary focus on the aluminum and bronze material.

All panels were coated with black and white Rust-Oleum High Heat spray paint which resists heat up to 650 °C. In order to facilitate full-field measurements, a random white speckle pattern was applied to the surface of each panel after the black base coat had dried through the application of a vinyl template. The final surface treatment resulted in a surface emissivity of 0.94.

The focus of this paper is experimental behavior. However, it is useful to have an idea of what to expect from a general theoretical standpoint, and the initial experimental goal of the investigation was motivated by designing a set-up to specifically exhibit interesting post-buckling behavior. Plate buckling is dependent on a variety of factors including geometry (aspect ratio, thickness), boundary conditions, material properties, etc. In order to reduce the sheer range of parametric variations in this study we largely focus on co-existing post-buckled equilibrium configurations in thin panels in which the aspect ratio is fixed at 2.71:1 (with typical length-to-thickness ratios on the order of 300–400). In all cases the two opposite short edges were clamped, with the longer edges both clamped and simply-supported. Given this aspect ratio we appeal to the classical results of thin-panel buckling in terms of aspect ratio as shown schematically in Fig. 2(a).

The details of this behavior depend on the boundary conditions, material, type of loading etc, but broadly speaking, we might expect a typical plate to elastically deform under axial loading in waves that produce buckled shapes that somewhat resemble harmonics, with the form depending on the aspect ratio. Mathematically, and with regard to Fig. 2(b) we might expect a deformed shape roughly of the form z(x,y)=A[1cos(2mπx/a)][(sin(πy/b)] in which the shape of the out-of-plane deflection z is characterized by m waves, for example, m = 3 in the long direction, with half sine waves in the short direction (for the case with simply-supported long edges). The specific form of the shape depends on a variety of factors (including subtle coupling and nonlinear issues), but given the aspect ratio we expect behavior that is most obviously related to m = 3 or 4, and in order to best represent this behavior (and avoid nodes) in bifurcation diagrams we shall focus attention on the deflection at the quarter and mid-points in the long direction as shown indicated in Fig. 2(c). Where indicated later, the 1/3 point is also used as a representative deflection point depending on the dominant participating mode.

For all testing conditions, the clamping fixture of interest was attached to a fixed (relatively massive) frame. All thermal loading was applied to the back side of each panel as demonstrated for the halogen lamp bank in Fig. 3(a) and for the inductive (localized) heater in Fig. 4(b). All images were captured on the front of the panel as the stereo 3D digital image correlation (3D-DIC) cameras and the forward looking infrared camera (FLIR) were mounted on the same tripod as shown in Fig. 3(b).

In the case of localized heating, a modified inductive heater was used to apply a very focused thermal loading over a relatively small area of the reverse side of the thin panels. A copper coil was designed and mounted behind the panel as shown in Fig. 4(a) and (b). The final coil induces a narrow line of eddy currents into the metal resulting in a narrow spot or line of heating. Both clamping configurations were mounted to the same static frame.

A quasi-static 3D-DIC system comprised of two Allied 6 MP (2752 pixels × 2200 pixels) CCD cameras were used to measure 3D coordinates of the thermally deformed panel. The cameras were fixed at a stand-off distance of 1.3 m at an angle of 18.4°. A 250 mm × 200 mm calibration target was used to calibrate the camera setup with 50 mm lenses, resulting in a calibration deviation of 0.018 pixels. Images were individually triggered with a shutter speed of 1/15 s.

The commercial digital image correlation software Aramis was used to compute the 3D coordinates at each deformed state. A subset size of 31 pixels × 31 pixels with a spacing of 21 pixels was selected for the analysis. The subset analysis parameters resulted in 4397 coordinate points across the surface of the panel. The computational area (green) and start points (red) for the full-field analysis are shown in Fig. 5.

The full-field temperature distribution was measured using a forward-looking infrared, or FLIR camera, calibrated with the measured surface emissivity. Typical (representative) temperature distributions for the ‘uniform’ heating cases and the local heating cases are shown in Fig. 6(a) and (b), respectively. The temperature scale in Fig. 6 is from blue (25 °C) to yellow (193 °C). It is apparent that the uniform heating cases will heat the panel and the frame while the local heating cases will have minimal effect on the frame. This is observed in Fig. 6, where the frame temperature is near 70 °C for the uniform heating case and 40 °C for the local heating case. Therefore, the difference between the maximum temperature on the panel and the average temperature (ΔT) at several locations along the frame is used to compare results from both heating cases. For all tests, the FLIR camera was triggered at the same time as the 3D-DIC system with a 1/15 s shutter speed.

Section snippets

Results

The testing results are reported primarily according to the boundary conditions (fully clamped and clamped-simply supported); type of material (bronze and aluminum); and thermal loading distribution (uniform and localized). However, we note that the aluminum panels could not be sufficiently heated with the inductive heater due to the electrical properties of aluminum. All data is collected over several heating and cooling cycles, and summarized in a single plot. We shall use the term ‘mode’ to

Strain calculation

An examination of the mechanical strain map provides more insight into the bulging associated with localized heating. It is important to first note that the strain calculated from digital image correlation coordinates [27] is a combination of thermal strain and mechanical strain as shown in Eq. (1):εTOTAL=εMECHANICAL+εTHERMAL.Therefore, the mechanical strain can be calculated by subtracting the total strain from the digital image correlation from the thermal strain determined from the

Conclusions

Thermal loading is perhaps one of the most common ways in which slender structural components (such as rectangular panels) are subject to excessive compressive axial lading. The suppression of axial expansion by a relatively massive support structure typically results in considerable out-of-plane deformation, since it is energetically easier for a panel to deform in this way. This type of instability behavior is characterized by considerable post-buckled stiffness, rather than complete

Data statement

The data is available upon request. Please contact the corresponding author.

Acknowledgment

The authors acknowledge the assistance of Brian T. Gockel and S. Michael Spottswood of the Structural Sciences Center who aided in the implementation of the inductive heater and measurement systems.

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