1、NATIONAL ADVISORY COMMITTEEFOR AERONAUTICSTECHNICAL NOTE 3783HANDBOOK OF STIRUCTUIRAL STABILITYPABT El - BUCKLING OF CURVED PLATES AND SHELLSBy George Gerard and Herbert BeckerNew York Universitys/WashinqtonAu_st 1957Provided by IHSNot for ResaleNo reproduction or networking permitted without licens
2、e from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Q TABLE OF CONTENTSPageSUMMARY . . iINTRODUCTION . iSYMBOLS 3PHYSICAL BEHAVIOR OF CURVED ELEMENTS . 8Correlation of Test Data and Linear Theory . 8Postbuckling Behavior 9STABILITY THEORY
3、OF CURVED ELEMENTS . iiLinear Stability Theory for Cylindrical Elements 12Boundary Conditions . . 16Solutions Based on Donnells Equation 17Case i. Axially compressed cylinders and curved plates ._“ 18Case 2. Cylinders under lateral and hydrostatic pressure . 20Nonlinear Stability Theory for Cylindri
4、cal Elements . 21Energy Criterion of Buckling . . 22CIRCULAR CYLINDERS UNDER AXIAL COMPRESSION . . 23Historical Background 24Buckling Behavior 25Long-Cylinder Range 26Transition Range 27Numerical Values of Buckling Stress . 28Plasticity-Reduction Factor . 29Effect of Internal Pressure .30CYLINDERS I
5、N BENDING . 31Historical Background . . 31Behavior of Circular Cylinders in Bending . 32Numerical Value of Buckling Stress for Circular Cylinders 34Behavior of Elliptic Cylinders in Bending 34Computation of Buckling Stress for Elliptic Cylinders . , 35Behavior of Circular-Arc Sections . 37Inelastic
6、Behavior of Long Circular Cylinders in Bending 38CIRCULAR CYLINDERS UNDER TORSION . . . . . . . 40Historical Background . . 40Experimental Data . . 41Buckling-Behavior of Cylinders Under Torsion 41Numerical Values of Torsional Buckling Stress 42Plasticity-Reduction Factors 43Effects of Internal Pres
7、sure . 44Elliptic and D-Section Cylinders 45iProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-PageCIRCULAR CYLINDERS UNDER EXTERNAL PRESSURE 46Historical Background . 46Test Data . 46Behavior of Cylinders . 47Buckling-Stress Equations . 48Radial press
8、ure . . 48Hydrostatic pressure . 48Effects of Plasticity . 49CIRCDLAR CYLINDERS UNDER COMBINED LOADS . 50Historical Background . 50Interaction Equations . 50Axial Compression and Bending . 50Axial Load and Torsion . _ 51Bending and Torsion 51Axial Compression, Bending, and Torsion 51Transverse Shear
9、 and Bending 52CURVED PLATES UNDER AXIAL COMPRESSION 53Historical Background . 53Summary of Test-Specimen Details 54Buckling Behavior of Axially Compressed Curved Plates . 54Initial Eccentricity 57Inelastic-Buckling Behavior 57Effect of Normal Pressure 58SPHERICAL PLATES UNDER EXTERNAL PRESSURE . 59
10、Historical Background . 59Initial Imperfections . 60Analysis of Initial-lmperfection Data 62Compressive-Buckling Coefficients 64Numerical Values of Buckling Stress 64Effects of Plasticity 65CURVED PLATES UNDER SHEAR 69Historical Background 65Test Data 66Behavior of Curved Plates Buckling Under Shear
11、 . 67Numerical Values of Buckling Stress 67Plasticity-Reduction Factors 68Effects of Internal Pressure . 68CURVEDPLATES m_OERCO_mD S_AR AND LONGr_INALCOMPRESSION 69iiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-PageAPPENDIX A - APPLICATION SECTION
12、 . 71Compressive Buckling 71Circular cylinders 71Elliptic cylinders . , . 72Curved plates 72Bending Buckling of Long Cylinders . 73Circular cylinders 73Elliptic cylinders . 73Torsional Buckling of Cylinders . 73Circular cylinders 73Elliptic cylinders and D-tubes 74Shear Buckling of Curved Plates 75B
13、uckling Under External Pressure 75Circular cylinders 75Spherical plates . 76Buckling Under Combined Loads 77Circular cylinders 77Curved plates 77REFERENCES . 78TABLES 85FIGURES 91iiiProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot f
14、or ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY C(_MITTEE FOR AERONAUTICSTEC ICALNCTE3783HANDBOOK OF STRUCTURAL STABILITYPART III - BUCKLING OF CURVED PLATES AND SHELLSBy George Gerard and Herbert BecketSU_ also, width of curvedplate, in.c chord of cir
15、cular-arc section, in.C compressive-buckling coefficient for long cylindersCb bending-buckling coefficient for long cylindersD bending rigidity, Et3/_2(1- v2_, in-lbd diameter of spherical plate (chord width)_ in.E elastic (Youngs) modulus, psiEs secant modulus, psiEt tangent modulus, psiF stress fu
16、nction for cylindersg exponent in expression for aoH depth of circular-arc section, in.K constant in expression for aokb buckling coefficient for cylinders in bendingkc buckling coefficient for axially loaded cylinders andsingly curved plateskp buckling coefficient for hydrostatic pressurekpl buckli
17、ng coefficient for flat plate, in generalks buckling coefficient for singly curved plate in shearkt buckling coefficient for cylinder or D-tube in torsionProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3783 5ky buckling coefficient for radial
18、 pressure on cylinderL length of cylinder or curved plate, in.L1,L2 wave length of buckle axially and circtmferentially asused in expression for ao, in.M bending moment, in-lbm wave number in axial direction of cylinders and singlycurved platesNx,Ny,Nxy axial, circumferential, and shear loads applie
19、d tocylindern wave number in circumferential direction of cylinders andsingly curved platesp pressure, psiRb stress ratio for bending on cylinderRc stress ratio for axial compression on cylinders and singlycurved platespressure ratio for cylinders and singly curved platesRs stress ratio for shear on
20、 singly curved platesRt stress ratio for torsion on cylindersRx stress ratio for axial loading, either tension or compres-sion, on singly curved plater radius, in.critical radius of curvature on section of ellipticcylinder in bendingS sensitivity factor in expression for aoSa section modulus of circ
21、umscribed circle, _a2tSc section modulus of circular cylinder, cu in.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 NACA TN 3783Se section modulus of elliptic cylinder, cu in.t sheet, plate, or cylinder-wall thickness, in.U,Uo unevenness factors i
22、n expressions for aou,v,w displacements in x-, y-, and z-directions, in.X dimensional factor in expression for ao, in.x,y,z coordinates for circular cylinders and singly curvedplates, axial, tangential, and radial directions,respectivelyy/a elliptic cylinder parameter (eq. (36)Z general length-range
23、 parameter for cylinders, singlycurved plates, and spherical platesZL = L2(I - Ve2)!/2/rt7 gradient factor7_ strain gradient factor7d stress gradient factorstrain, in./in.plasticity-reduction factorbuckle wavelength, in.Provided by IHSNot for ResaleNo reproduction or networking permitted without lic
24、ense from IHS-,-,-NACA TN 3783 7_ magnification factor, kexp/kempv Poissonsratio,Vp- (v_-Ve)(_slE)Ye elastic Poissons ratio, 0.3 in this reportVp plastic Poissons ratio, generally 0.5p shape factor for inelastic-bending-stress distributionnormal stress, psi_b actual plastic stress at extreme fiber o
25、f cylinder inbending_cl classical buckling stress of sphere under external pressure_i = (_x2 + _y2 - _x_y + 3T2)I/2_r bending modulus of rupture, M/scT shear stress, psie cylindrical coordinateX curvatureSubscripts:cr critical (buckling stress)emp empiricalexp experimentale edge; also, elliptic cyli
26、ndero initialb bendingc compression; also, circular cylinderx,y in axial and tangential directions, respectivelyProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACA TN 3783PHYSICAL BEHAVIOR OF CURVED ELEMENTSCorrelation of Test Data and Linear Theo
27、ryIn Part 1 of this Handbook (ref. l) the buckling of flat plates wasreviewed. The close correlation of experimental data on the elastic andplastic buckling of flat plates under various types of loadings andboundary conditions confirms the use of classical linear stability conceptsin such problems.
28、Furthermore, it suggests that small initial imperfec-tions unavoidably present in practical structural elements are unimpor-tant from an engineering standpoint.In investigating the elastic buckling of thin-wall circular cylinders,curved plates, and thin-wall spheres, classical stability theory has b
29、eenused also. In general, however, the close correlation between theory andtest data observed for flat plates is not obtained for curved elements.The amount of agreement varies and depends upon the type of loading andthe geometric parameters of the c_rved element.The most complete test data are avai
30、lable for cylinders. These datawere reviewed by Batdorf (ref. 3) and were compared with a simplifiedlinear buckling analysis based on the use of Donnells equations. This _set of equations as well as others are discussed in the section entitled“Stability Theory of Curved Elements. “ For the purposes
31、here, it willsuffice to compare the results of the simplified analysis with availabletest data.Representative elastic-buckling data for cylinders under axial com-pression, torsion, and lateral pressure are shown in figure i. It canbe observed that for compressive loading the best test data at failur
32、eare approximately one-half of the theoretical buckling values with somedata as low as i0 percent of theory.Furthermore, the scatter in the data is large, even on the logarithmicplots on which the results are shown because of the large numerical rangeof the parameters. Other test data on elastic buc
33、kling of curved platesunder axial compression, spheres under hydrostatic pressure, and cylindersunder bending all behave in the characteristic manner of axially compres-sed cylinders.For torsion loads the test data on failure of the cylinders are inconsiderably better agreement with buckling theory
34、than are those forcompression. Here too, however, the test data are consistently belowthe theoretical values. In the case of buckling under lateral pressure,the relatively small amount of test data is in good agreement with theory.The particularly poor agreement between linear theory and tests forax
35、ially compressed curved elements has motivated considerable theoreticalProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3783 9investigation to determine the cause of such behavior. Some investi-gators have maintained that such elements are par
36、ticularly sensitive toinitial imperfections which lead to premature failure. Others haveabandoned classical buckling concepts. By use of large-deflection theoryin conjunction with deflection functions corresponding to the experi-mentally observed dismond pattern, it was found that neighboring large-
37、deflection equilibrium configurations exist at loads less than those ofthe linear theory. It has been suggested that the small amount of energyrequired to trigger the jump to the neighboring equilibrium configura-tions can be supplied by small vibrations in the testing machine. Thus,the compressed c
38、ylinder cannot reach the classical load and fails at afraction of this value.These approaches are discussed at some length in the sections“Stability Theory of Curved Elements“ and “Circular Cylinders UnderAxial Compression.“ At this point, however, it seems important to inquirefor the reasons for th
39、e apparent failure of linear theory for compressivebuckling of curved elements. In this case, large-deflection theory mustbe introduced, whereas for torsional buckling linear theory providesreasonable agreement with test data and for cylinders under lateral pres-sure good agreement is obtained.Postb
40、uckling BehaviorSome explanation on physical grounds is required to indicate whenlarge-deflection effects may assume importance in particular bucklingproblems. For such an explanation, it is logical to consider the post-buckling behavior of various elements, since this is the region of largedeflecti
41、ons.A schematic representation of the postbuckling behavior of axiallycompressed colunuis, flat plates, aud cylinders is shown in figure 2 forboth theoretically perfect elements and those containing initial imper-fections. It is assumed that all elements behave elastically.For the perfect column, th
42、e postbuckling behavior is essentiallyhorizontal in the range of Wave depth/Shell thickness values consid-ered here (elastic effects are negligible) and buckling can follow eitherthe right branch (0, i, A+) or the left (0, i, A-). The horizontalbehavior can be attributed to the fact that, after buck
43、ling, no signifi-cant transverse-tension membrane stresses are developed to restrain thelateral motion and, therefore, the column is free to deflect laterallyunder the critical load.The flat plate, however, does develop significant transverse-tensionmembrane stresses after buckling because of the re
44、straint provided by theboundary conditions at the unloaded edges. These membrane stresses actProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-i0 NACA TN 3783to restrain lateral motion and thus the flat plate is capable of carryingloads beyond buckling
45、 as indicated by the approximately parabolic char-acter of the stress-deflection plot of figure 2. The flat plate alsocan follow either the right branch (0, i, h+) or the left (0, i B-).For the axially compressed curved plate, the effect of the curvatureis to translate the flat-plate postbucklingpar
46、abola downward and towardthe right, depending upon a width-radius parameter. For the completelong cylinder a considerable translation occurs. Note that by shiftingthe parabola to the right buckling would tend to follow the right branchonly (0, i, C) because of the lower loads involved, with the resu
47、lt thatthe inward type of buckling is observed for curved plates and cylinders.This inward buckling causes superimposed transverse membrane stressesof a compressive nature so that the buckle form itself is unstable.As a consequence of the compressive membrane stresses 3 buckling ofan axially compres
48、sed cylinder is coincident with failure and occurssuddenly (snap buckling, “oilcanning“) accompanied by _2w (4)Ny-+p=Or2_2The plasticity coefficients are defined as follows:Am = 1 - 4A3 = l-_m 2A21 = AI2 = i - _dxd 2A31 : AI3 = _x TA32 = A23 = _dyTProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3783 15where :The axial rigidity is:B = 4_,st/3 (5)The bending rigidity is:D = Est3/9 (6)In the elastic region, _ :
