1、NASA CR-132548 SHEBR BUCKLING OF SQW PEBPOBBTBD PLATES John F. Grosskurth, Jr., Rlchard bl. White, Richard E. Galaher, and Gareth B. Thomas ( N A SA -C Po 1 3 2 5 4 3) SGUAEE PEFFC3ALi:C PLkiiS (Corcei,. Cniv.) 5 11 E k 3 E ,i CK L I N G ,J 47 p HC 13.35 CSCL 20K 375- lLdO4 Prepared under Grant No.
2、NGL 33-010-070 by CORNELL UNIVERSITY Ithaca, 14y 14850 for NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ABSTRACT The behavior of thin square perforated plates under the action of uniform shear deformat
3、ion is studied experi- mentally and analytically using finite element analysis. Elastic Shear buckling strength is established as a function of the diameter of a round, centrally located hole in the plate. Post buckling behavior and the behavior of perfor- ated plates with various ring stiffeners ar
4、e also studied experimentally. Key Words Shear buckling Plate buckling Centrally located circular perforations in plates Finite element analysis Experimental study Re I nf orc emen t Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot
5、for ResaleNo reproduction or networking permitted without license from IHS-,-,-TABLE OF CONTENTS List of Symbols List of Figures List of Tables 1. Introduction 2. 3. Analysis Test Specimens and Experimental Program 4. 5. Concluding Remarks References Results and Analysis - Test Comparisons Figures T
6、ables Page V vii viii 1 4 9 15 21 23 24 42 iii Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-b d k t U v W E N T v x V T LIST OF SYMBOLS shorte? side
7、length of rectangular plate diameter of hole buckling coefficient plate thickness x direction displacement component y direction displacement component z direction displacement component modulus of elasticity linear stiffness matrix unitless buckling parameter geometric stiffness matrix diagonal ten
8、sion force shear force lateral displacement load intensity factor Poissons ratio normal stress shear stress nodal displas?2ments nodal displacements at unit load intensity V Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for Resal
9、eNo reproduction or networking permitted without license from IHS-,-,-List of Figures 1. Geometry of Test Specimen. 2. Reinforcement Patterns Around Perforat ions in Plates with d/b = 0.4. 3. Testing Apparatus. 4. Exploded View of Test Fixture. 5. Unperforated Test Specimen. 6. Plate Element Nodes a
10、nd Degrees- 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. f -Fr edom. Mode of Deformation of Specimen Due to Imposed Edge Displacements. Finite Element Gridworks. Finite Element Gridworks for Convergence Study of Shear Buckling Calculations for Unperforated Plate. (See Fig. 8a for 36-Element Grid
11、work). Comparison of Various Finite Element Models with Classical Solution. Comparison of Solutions for T, as a Function of d/b Ratio. Unstiffened Plates. Pre-buckling In-plane Stiffness as Function of Various d/b Ratios. Unstiffened Plates. Contour Plots of Normal Modal Displacements for Various d/
12、b Ratios T vs. dlat for Four Unstiffened Plates. Criteria for Defining Critical Buckling Load Applied to Test Data for Unperforated Plate. Buckling and Ultimate Loads Determined from Test Results for Various d/b Ratios. Post-buckling Behavior of Plate with Uniform Edge Shear Deformation with d/b = 0
13、.6. Specimens after Testing. Comparison of Experimental and Analytical Buck- ling Loads for Various d/b Ratios. Page 24 24 25 26 27 28 28 29 30 30 31 32 33 34 35 36 37 38 39 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-20. T vs. 61at for Equal Cro
14、ss-Section Stiffeners. 40 21. T vs. dIat for Equal Volume Stiffeners. 41 List of Tables 1. Lmparison of Analyss and Test Strains at Selected Gage Locations. 42 2. Inflection Point Buckling Loads for Both Unper- foreted Plate and Stiffened and Unstiffened Plates with 4 -Inch Perforat ion. 42 viii Pro
15、vided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1. INTRODUCTION Thin plates occur in a wide variety of structures. While a good deal of our early knowledge about the behavior of thin plates has evolved from research directed to civil engineering structu
16、res such as plate girders, new applica- tions in aerospace structures and pressure vessels have ac- celerated research in recent years. access to some other portion of the parent structure. Per- forations are made in plates to allow permamnt passage of electrical, hydraulic, or other conduits, or si
17、mply to per- mit easy access for painting or servicing. Where a reduc- tion in strength is permissible, a hole might be placed in a plate to reduce weight. This is common in aerospace structures. In some cases, the reduction in the strength of the plate can be offset by the presence of a stiffener.
18、Plate elements are often subjected to complex loading conditions. To understand the behavior of perforated plates under complex loads, we must first know something of their behavior under more simple loading conditions. The behavior of thin, square, perforated plates subjected to in-plane shear is s
19、tudied experimentally and analytically in this report. The plates have clamped edges and centrally located, circular perforations. An experimental study of stiffened perforated plates is also presented here. A key aspect in the behavior of any thin plate is its buckling strength. Thin plates commonl
20、y become unstable at stress levels far below the proportional ;iml.t of the material. In some cases, the lmge transve-sz deformations associated with buckling can be tolerated an: only the ultl- mate strength of +bo plate is of any practical consequence. Elsewhere, aerodynamic or aesthetic considera
21、tions dictate that buckling cannot be allowed. Whatever the application, plate elements often block Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Plate buckling strength is usually expressed in terms of the critical stress, aCr or fCr. The critical
22、 stress is the stress which, when applied uniformly along the edges of the plate, produces instability. For an isotropic square plate of thickness t, width b, elastic modulus E, and Poissons ratio u, this stress can be expressed as n2 Et2 22 =k 12 ( 1-u )b cr, Tcr U where k is known as the stability
23、 coefficient. Stability coefficients for compression and shear buckling of plates with various aspect ratios and edge conditions have been calculated and can be found in Reference 1. For the case of shear buckling of a square plate with clamped edges, this coefficient has been determined by Budiansk
24、y and Connor (Ref. 2) to be 14.71. Perforated plates, however, present some analytical problems. The case of the square perforated plate under edge compression has been studied quite extensively since 1947. The most important works in this period are those of Levy, et al. (Ref. 3), Schlacr .Ref. 4 a
25、nd 5), Kawai and Ohtsubo (Ref. 61, 2nd Vann and Vos (Hef. 7). The analytical methods used in these papers range froin energy methods and numerical integration in the report of Levy, et al. to finite element analysis as used by Vann and Vos. The shove work excluded consideration of shear buckling. In
26、finitely long shear webs have received some attention, but their relevance to the square pla%e is minimal and they will not be discussed in this report. The first serious attempt at predicting the critical shear stress of square perforated plates was made by Kroll (Ref. 8) in 1949. Using Timoshenkos
27、 energy method, which calls for the evaluation of two double surface integrals, Kroll obtained theoretical buckling coefficients for simply supported square plates with circular holes having diameters (d) equal to 1/8 and 1/4 the length of a side. Provided by IHSNot for ResaleNo reproduction or netw
28、orking permitted without license from IHS-,-,-Numerical integration was used, and 22 points were needed in each octant to reduce the error for the known unpierced case to less than 5%. Analysis of the perforated plates indicated that the smaller hole reduced the coef- ficient by only 0.2%, while the
29、 larger hole cause1 a 22.6% reduction in the coefficient. Hole reinforcements were also analyzed and were found to increase the coefficient by as much as 334% over that for an unreinforced hole. In 1967, Rockey, Anderson, and Cheung (Ref. 4) solved the same problem using finite element analysis. Wit
30、h an idealization containing 56 triangular elements in the ana- lyzed quadrant, the solution for the unperforated case dif- fered from classical theory by 9.7%. Solutions for the per- forated cases differed markedly from those obtained by Kroll. Four hole sizes were analyzed with simply supported an
31、d clamped edges. For the simply supported plate with a hole diameter to plate width ratio d/b = 0.125, Rockey, et al. found a reduction in the buckling coefficient of about 15%, whereas Kroll reported only a 0.2% reduction. The two papers were in somewhat better agreement with respect to perforated
32、plates with a hole diameter to plate width ratio d/b = 0.250, for which Rockey, et al. observed a 28% reduction in the buck- ling coefficient, compared to Krolls 22.6%. Rockey, et al, investigated hole reinforcements, like Kroll, and found them extremely effective in canceling losses in buckling str
33、ength due to perforations. An experimental program was conducted and agreement with their analysis was good for the smaller diameter holes. Agreement was not as good for the larger holes. Both Kroll and Rockey, et al, studied uniform applied shear stress rather than the case encountered in testing,
34、that of uniform shear deformation. This was most graphical- ly apparent in the investigation of Rockey, et al., wherein uniform stress was analyzed and uniform deformation tested. While this distinction has littie effect on the classical Provided by IHSNot for ResaleNo reproduction or networking per
35、mitted without license from IHS-,-,-4 case of the unperforated plate, it has definite consequences when a hole is introduced. As the diameter of the hole in- creases, the stiffness of the plate continues to change along the edge and the two cases become more dissimilar. This distinction was observed
36、 for the perforated plate in compression by Schlack in 1964. It was taken into ac- count in his analysis, which used the Rayleigh-Ritz energy method, and was borne out by his testing. The study 1-eported here is restricted to perforated plates subjec:ed to uniform shear deformation. The choice of te
37、st specimen geometries and the experimental program are described in Section 2. An analytical approach to elastic stability of unstiffened perforated plates is pre- sented in Section 3, along with a summary of computed re- sults. The experimental results for both unstiffened and stiffened platec 92
38、g;iven in Section 4 and comparisons between analysis and test results are made. The final sec- tion of the report contains the concluding remarks. - 2. TEST SPECIMENS AND EXPERIMENTAL PROGRAM Specimen Size and Materials: The experimental program was conC.cted on 10“ square aluminum plates with a thi
39、ckness of 0.063“. The 10“ size was chosen for convenience. It was felt that plates of this size would be large enough for careful observation and instrumentatlon, and yet not too large to test in a standard testing machine. Larger plates would also have necessitated a larger and more costly test fix
40、ture. The material and thickness of the plates were chosen to ensure elastic buckling. Since very thin plates would be difficult to handle and would probably have initial im- perfections of more significance than thicker plates, the latter were considered more desirable than the former. Since buckli
41、ng stress is proportional to the square of the Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-5 plate thickness and to the elastic modulus, thicker plates could only be used with lower modulus materials. Thus, aluminum, with an elastic modulus appro
42、ximately one third that of steel, was chosen. Using Eq. 1, it was found that aluminum plates with a thickness of 0.063“ and a side length of 10“ would have a critical stress of 5.4 ksi. This is well below the yield stress and proportional limit of most of the common struc- tural aluminum alloys. The
43、 specific alloy chosen was 6061-T6, which has a relatively high yield point, an extensive linear post-yield region in its stress-strain curve, and a relatively small difference between Its yield stress and ultimate strength. Hence, it approaches the elastic-plastic properties of structural steel and
44、 was chosen on that basis. Six tensile coupons were tested to evaluate the signifi- cant mechanical properties of the material. The properties of interest are the initial modulus of elasticity E, Poissonts ratio v, the proportional limit Q the yielrl stress (T and the ultimate strength uu. as the po
45、int at which tensile strain first deviated from linearity by 0.0001 and the 0.2% offset approach was used to define the yield point. The above properties, which did not vary significantly with the direction in which the material v = 0.3, (Tp - 34,000 6 was rolled, were E = 10.2 x 10 P.s., p.s.i., Q
46、= 30,000 p.s.i. and aU = 45,000 p.s.i. Three groups of specimens were tested. The first group were simply unperforated plates, (Figure 1, d = 0), used to establish “benchmark“ values for the buckling loads. The second group of specimens was used to study the influence of unstiffened hole size on she
47、ar buckling capacity. Central holes with diameters (d) of 2, 4, and 6 j.n. were cut in these specimens, giving perforated plates with d/b ratios of 0.2, 0.4, and 0.6. P Y The proportional limit was taken Y Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-
48、,-,-6 The third group of specimens (Fig. 2) consisted of perforated plates with stiffening material added around the perforations for the purpose of increasing the buckling load up to or beyond the buckling load of the unperforated plates. Five stiffened plates with four-inch holes were testcd. All five stiffeners were flat rings made of Lie sane material as the basic plates. Choice of Stiffeners There are several rules encountered in prrlctice WhiCki can be used to determine
