The Effects of Plate-Support Condition on Buckling Strength of Rectangular Perforated Plates under Linearly Varying In-Plane Normal Load
— Mechanical buckling analysis of rectangular plates with central circular cutout is performed in this paper. The finite-element method is used to study the effects of plate-support conditions, aspect ratio, and hole size on the mechanical buckling strength of the perforated plates subjected to linearly varying loading. Results show that increasing the hole size does not necessarily reduce the mechanical buckling strength of the perforated plates. It is also concluded that the clamped boundary condition increases the mechanical buckling strength of the perforated plates more than the simply-supported boundary condition and the free boundary conditions enhance the mechanical buckling strength of the perforated plates more effectively than the fixed boundary conditions. Furthermore, for the bending cases, the critical buckling load of perforated plates with free edges is less than perforated plates with fixed edges.
Mechanical- and Thermal-Buckling Behavior of Rectangular Plates With Different Central Cutouts
Mechanicaland thermal-buckling analyses were performed on rectangular plates with central cutouts. The cutouts were either circular holes or square holes. The finite-element structural analysis method was used to study the effects of plate-support conditions, plate aspect ratio, hole geometry, and hole size on the mechanicaland thermal-buckling strengths of the perforated plates. By increasing the hole size, thermal-buckling strengths of the plates could be enhanced. The compressive-buckling strengths of the plates could also be increased considerably only under certain boundary conditions and aspect ratios. The plate-buckling mode can be symmetrical or antisymmetrical, depending on the plate boundary conditions, aspect ratio, and the hole size. For the same cutout areas (i.e., same plate weight density), the buckling strengths of the same-sized plates with square holes generally surpass those of the plates with circular holes over the range of hole sizes. The results and illustrations provide vital information for the efficient design of aerospace structural panels. NOMENCLATURE c side of square hole, in. d diameter of circular hole, in. E Young’s modulus, lb/in E33 triangular combined membrane and bending element E43 quadrilateral combined membrane and bending element G shear modulus, lb/in JLOC joint location (or grid point or node) l length of rectangular plates, in. compressive force intensity in y-direction, lb/in. SPAR structural performance and resizing finite-element computer program T temperature, °F assumed temperature for material, °F room temperature, = 70 °F t thickness of plates, in. w width of plates, in. x, y rectangular Cartesian coordinates α coefficient of thermal expansion, in/in-°F ∆T temperature increase, °F ν Poisson’s ratio Subscripts ( )cr critical value at buckling ( )n n th iteration Ny Ta Tr Tr n 1 2 3... , , = ( ) INTRODUCTION In aerospace structures, cutouts are commonly used as access ports for mechanical and electrical systems, or simply to reduce weight. Structural panels with cutouts often experience compressive loads that are induced either mechanically or thermally, and can result in panel buckling. Thus, the buckling behavior of those structural panels with cutouts must be fully understood in the structural design. For an unperforated rectangular plate of finite extent (i.e., with finite length and finite width) under uniform compression, the closed-form buckling solutions are easily obtained because the prebuckling stress field is uniform everywhere in the plate. When a finite rectangular plate is perforated with a central cutout (e.g., a circular or square hole), however, the buckling analysis becomes extremely cumbersome because the cutout introduces a load-free boundary that causes the stress field in the perforated plate to be nonuniform. Hence, the closed-form buckling solutions are practically unobtainable, and various approximate methods had to be developed to analyze such perforated plates. The buckling of flat square plates with central circular holes under in-plane edge compression has been studied both theoretically and experimentally by various authors (refs. 1–12). The methods of theoretical analysis used by most of the past investigators (refs. 1–3, 5) were the Rayleigh-Ritz minimum energy method and the Timoshenko method (ref. 13). However, except for Schlack (ref. 3) and Kawai and Ohtsubo (ref. 5), the theoretical analysis methods used do not allow the boundary and loading conditions to be precisely defined for larger hole sizes because the stress distributions of the infinite perforated plate are used as the prebuckling stress solution for the finite perforated plate. Thus, most of the earlier buckling solutions are limited to small hole sizes, and are not fit for studying the effects of different plate boundary conditions on the buckling strengths of the finite plates with arbitrarily sized holes using those approximate solutions. Using the Rayleigh-Ritz method, Schlack (ref. 3) analyzed the buckling behavior of a simplysupported square plate with a circular hole, subjected to uniform edge displacements with three arbitrary displacement functions, and calculated the buckling displacements. The buckling loads were then calculated using the stress-strain relationships. Ritchie and Rhodes (ref. 7) studied the buckling behavior of both square and rectangular simply-supported perforated plates. Their theoretical analysis employed an approximate approach using a combination of Rayleigh-Ritz and finite-element methods. These methods are reasonably accurate for small holes, but lose accuracy when dealing with larger holes. The results of the analysis show that the buckling behavior of perforated rectangular plates is quite different from that of perforated square plates, and that the buckling mode is dependent on the hole size. Kawai and Ohtsubo (ref. 5) also studied the perforated square plates using the Rayleigh-Ritz procedure with the prebuckling stress distribution determined by the finite-element method. To reduce the labor of numerical calculations, the double integrations in the energy procedure for each finite element were transformed into line integrals around the element boundary using the well-known Gauss theorem. To minimize the mathematical complexities, Nemeth (refs. 8–11) analyzed perforated square orthotropic plates by converting the classical two-dimensional buckling analysis into an equivalent onedimensional analysis by approximating the plate displacements with kinematically admissible series. In the analysis, the two unloaded edges were assumed simply supported, and the loaded edges were either simply supported or clamped. This approximate buckling analysis predicted the buckling loads to within 10 percent of those calculated using the finite-element method. Nemeth’s analytical and experimental results (ref. 10) indicated that increasing the hole size in a given plate does not always reduce the 2 buckling load. Using the finite-element method, Lee et al. (ref. 12) examined the buckling behavior of a square plate with a central circular hole. The studies, however, were limited to small hole sizes. As mentioned earlier, most past theoretical analyses of perforated plates considered mainly the square plates under simply-supported boundary conditions to minimize the mathematical complexities. Likewise, most past experimental studies on perforated plates were limited to square plates under simplysupported boundary conditions because of the need to compare the test data with the existing theoretical results. The past investigations, in general, lack comprehensive information on the buckling behavior of rectangular plates of arbitrary aspect ratios, containing holes of arbitrary sizes, and supported under any specified boundary conditions. With the availability of powerful tools such as the well-developed, finite-element structural analysis computer programs, it is now possible to calculate the prebuckling stress fields and the buckling eigenvalue solutions quite accurately for the finite rectangular plates of any aspect ratios, containing cutouts of any geometry and any hole sizes, under any specified boundary and loading conditions. This report investigates the mechanicaland thermal-buckling analyses of rectangular plates containing arbitrarily-sized central circular holes or square holes. A finite-element method was used to study the effects of plate aspect ratio, hole geometry, hole size, and plate boundary conditions on the mechanicaland thermal-buckling strengths of perforated plates. DESCRIPTION OF THE PROBLEM The geometry of the perforated rectangular plates and different boundary conditions used in the finite-element analysis are described as follows. Geometry Figure 1 shows the geometry of two types of perforated rectangular plates with length l, width w, and thickness t. The central cutout is either a circular hole with diameter d (fig. 1(a)), or a central square hole with side c (fig. 1(b)). Table 1 lists the dimensions of various perforated rectangular plates analyzed. Notice that all the plates have the same width, w = 20 in., and the same thickness, t = 0.1 in. In table 1, the range 0 ~ 0.7 covers 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7. For a given plate aspect ratio, the 15 hole-size cases will provide 15 data points for plotting each buckling curve in the ( )cr vs. d /w (or c /w) plots. If the buckling curve has any sharp bends in certain regions, however, additional data points for d /w (or c /w) values not listed in table 1 were generated to define those sharp bend regions more accurately. Table 1. Dimensions of perforated plates. w, in. t, in. l /w d /w c /w 20 0.1 1.0 0 ~ 0.7 0 ~ 0.7 20 0.1 1.5 0 ~ 0.7 0 ~ 0.7 20 0.1 2.0 0 ~ 0.7 0 ~ 0.7 Ny
Experimental critical loads for perforated square plates
Experimentally determined critical loads are presented for simply supported square plates with a circular central hole under the action of uniform edge displacements for hole diameters varying up to seventenths the dimension of the plate. The critical load is defined as the inflection point on the load-deflection curve as measured by a set of dial gages at the edge of the hole. Least-squares-curve fitting techniques are used for reducing all experimental data and the entire set of computations is carried out on the electronic digital computer. Finally, it is shown that the experimental results agree well with the theoretical critical loads as determined by the Ritz energy method.