It is well established that certain structural buckling problems are extremely sensitive to small changes in configuration: geometric imperfections, load application, symmetry, boundary conditions, etc. This paper considers the behavior of a very shallow arch under lateral point loading, and specifically under the influence of changes in the thermal environment. In some ways the system under study is especially sensitive since small changes influence whether the arch ‘snaps-through’ or not. The experimental results provide insight into the challenges of understanding the behavior of these types of structural components in a practical, and thus necessarily imperfect, situation. The focus is on static loading or at least quasi-static loading, in which loading occurs on a slow time scale. This study also acts as a back-drop for studying the dynamic behavior of shallow arches, an area of concern in the context of aerospace structural components.
The shallow arch provides an important paradigm in non-linear structural behavior. It is also quite representative of a broad class of curved, slender members used throughout aerospace structures. Although the geometry of a very shallow arch is not particularly different from that of a flat beam, the response to transverse loading may be quite different. The presence of even small amounts of curvature allows for strongly non-linear behavior including snap-through buckling, i.e., the sudden dynamic jump from one equilibrium configuration to a remote (co-existing) configuration that is often associated with an inverted position. Furthermore, the geometric ’depth’ of the arch, measured in terms of curvature or the rise/span ratio, may lead to an asymmetric loss of stability, even in those cases where the structure and loading are nominally symmetric.
This paper will focus attention on a specific arch: a thin steel strip whose (unstressed) equilibrium configuration includes a slight curvature, clamped at both ends (i.e., zero displacement and velocity boundary conditions for nodes on each end), subject to transverse point loading, and also under elevated thermal loading conditions. The focus of the research was to assess the structural response relative to imperfections in shape, clamping force, load location, and thermal conditions, i.e., a sensitivity study.
The motivation for this paper arose from experiences at the Air Force Research Laboratory (AFRL) at Wright-Patterson Air Force Base (WPAFB) in Dayton, Ohio, where shallow arches had been the subject of various tests mostly focused on their dynamic response, for example, modal analysis to extract natural frequencies and mode shapes, and a statistical analysis of persistent and intermittent dynamic snap-through behavior of the beam subject to high-frequency excitation. The sensitivity and inherent difficulty associated with repeatability of the load-deflection behavior of these arches had been noted despite the relatively careful testing conditions. This paper will explore the sensitivity of such structures within an (inevitably imperfect or noisy) experimental context. These results will also be discussed in relation to a finite element (FEA) study in which an attempt is made to match the behavior observed in the laboratory.
By way of introduction, consider a shallow, clamped arch as shown in Fig. 1(a). When subject to a point load located at the center of the span, the arch deflects and may snap-through. If the arch is sufficiently shallow (the sketch has an exaggerated vertical scale for clarity), with symmetric loading and boundary conditions, the deflection will typically be symmetric, and on removal of the load the arch may or may not return to its original configuration. Fig. 1(c) shows a representative set of data taken from the lab, used here for illustrative purposes: details related to the systematic experimental study will be given later. The force, F, was applied via a displacement-controlled device that pushed on the arch. The force (measured by a load cell) increased and then decreased until losing contact as the arch suddenly snapped to its inverted configuration, which is indicated by the lowest dashed curve in part (a). The loading mechanism was then re-contacted and further data were taken, and a reversal in the direction of loading (shown in red) until contact was lost again. This behavior is typical of shallow arches.
It is shown that the static response of a light shallow arch is very sensitive to thermal and prestress loads, as well as the point of the application of transverse loads. These sensitivities make it very difficult both to achieve repeatable results in any set of experiments and to match experimental results with numerical modeling. Other sensitivities are inevitably related to initial geometry (including asymmetry) and clamping force. The sensitivities do, however, highlight the importance of obtaining knowledge of all the forces in real structures of this type, such as aircraft panels.
A secondary bifurcation in which the unloaded snapped-through configuration stabilizes under varying thermal conditions was investigated. It is argued that this condition may be of concern (along with the initial limit point bifurcation itself) in the design of curved structures, as it increases the likelihood of persistent large deformations under both static and dynamic loads.
