1、!L1LL=NATIONAL ADVISORY COMMITTEEFOR AERONAUTICSTECHNICAL NOTE 1928CRITICAL COMBINATIONS OF SHEAR AND DIRECT AXIALSTRESS FOR CURVED RECTANGULAR PANELSBy Murry Schildcrout and Manuel SteinLangley Aeronautical LaboratoryLangley Air Force Base, Va._.r _“ . _,_, r T_,-_ _ _TWashingtonAugust 1949!Provide
2、d by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2_ .L_IiProvided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS -,-,-NATIONAL ADVISORY C0_ FOR AERONAUTICSTECHNICAL NOTE 1928CRITICAL COMBINATIONS OF SHEAR AND DIRECT AXI
3、ALSTRESS FOR CURVED RECTANGULAR PANELSBy Murry Schildcrout and Manuel SteinSUMMARYA solution is presented for the problem of the buckling of curvedrectai_ular panels subjected to “combined shear and direct axial stress.Charts giving theoretical critical combinations of shear and direct axialstress a
4、re presented for panels having five different length-widthratios.Because the actual critical compressive stress of rectangularpanels having substantial curvature is known to be much lower thanthe theoretical value, a semiempirlcal method of analysis of curvedpanels suoJected to combined shear and di
5、rect axial stres_ is presentedfor use in design.INTRODUCT IONAn investigation was made to determine the combinations of shearand direct axial stress that cause simply supported curved rectangalarpanels to buckle. Because panels having substantial curvature areIcnown to buckle in compression at a str
6、ess well below the theoreticalvalue, the solution must be at least partly empirical. In order toeliminate the necessity for an extensive test program, a theoreticalsolution to the problem is presented and Js modified for use in design.The modifications to the theoretical interaction c_ves are basedu
7、pon results of tests on the buckling of curved rectangular panelsunder combined shear and axial compression and incorporate resultsfor curved panels subjected to shear alone (reference l) and axialcompression alone (references 2 and 3). The resulting empiricalinteraction curves are expected to give
8、a good approximation to theactual critical combinations of shear and direct axial stress.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACATN 1928abm,n,p,qrtwXYDEZamn, apqk sk x(Rs )th(Rs )expSYMBOLSaxial or circumferential dimension of panel, wh
9、icheveris largeraxial or circumferential dimension of panel, whicheveris smallerintegersradius of curvature of panelthickness of paneldisplacement of point on shell median surface inradial direction; positive outwardaxial coordinate of panelcircumferential coordinate of panelflexural stiffness f pan
10、el per unit length IL _1t-3-2_2Young s modulus of elasticityb2 2curvature parameter (_K_)coefficients of terms in deflection functionscritical-axial-stress coefficient kW)theoretical shear-stress ratio (ratio of shear stresspresent to theoretical critical shear stress inabsence of other stresses)exp
11、erimental shear-stress ratio (ratio of shear stresspresent to experimental critical shear stress inabsence of other stresses)IProvided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS -,-,-NACA TN 1928 3(Rx) th(Rx) exp._ =abT(Ix_4_Ttl =-.-_+ 2theoretical direct-e
12、_ial-stress ratio (ratio of directaxial stress present to theoretical direct axialstress in absence of other stresses)experimental dlrect-axial-stress ratio (ratio of directaxial stress present to experimental critical directaxial stress in absence of other stresses)Poisson s ratiocritical shear str
13、esscritical axial stress_4 _4ax2 y2RESULTS AND DISCDSSIONTheoretical SolutionThe combinations of shear and axial stress that cause rectangularcurved panels (fig. l) to buckle may be obtained from the equationsks_2DT=b2tandkx_2D_x = b2 twhen the stress coefficiente ks and kx are known. The theoretica
14、lcombinations of stress coefficients for simply supported curvedrectangular panels having different ratios of circumferential toaxial dimension are given in figure 2. These combinations of stresswere obtained from the solution presented in the appendix, basedupon the small deflection theory of elast
15、ic stability of curvedProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 1928plates. Each part of figure 2 presents results for panels having aconstant ratio of circumferential to axial dimsnsion but various valuesof the curvature parameter Z.Fig
16、ure 2 indicates that, irrespective of the length-_idth ratioof the panel, the theoretical interaction curves approach those for acylinder as the value of Z increases. The value of Z at which thiscorrelation between the cylinder and rectangular panel becomss closeincreases as the ratio of circumferen
17、tial to axial dimension of thepanel decreases. The critical compressive stress of rectangular panelsis very nearly equal to that of cylinders even at low values of Z,whereas the critical shear stress differs greatly at low values of Z;therefore, a good indication of the point at which the interactio
18、n curvefor a panel is approximated very closely by that for a cylinder is thevalue of Z at which the respective critical she_r stresses are nearlyequal. These values of Z m_ybe obtained for simply supported panelshaving various length-width ratios from figure 3, which is taken fromreference 1. At su
19、fficiently high values of Z, as in the case of acylinder (see reference 4), the theoretical interaction curves instress-ratio form maytherefore be approximated in the compressionrange by a straight line from (Rx)th = 1 to (Rs)th = 1 and in thebeginning of the tension range by a straight line of slop
20、s -0.8passing through (Rs)th = 1. The critical-axial-stress and critical-shear-stress coefficients are obtained, respectively, by multi-plying the stress ratio (Rx)th by the theoretical critical stressesfor axial compression alone and the corresponding stress ratio (Rs)thby the theoretical critical
21、stresses for shear alone. These theoreticalcritical stresses maybe obtained from figures 4 to 7, which are takenfrom references 1 to 3-Although a theoretical solution is given only for simplysupported panels, the conclusions drawn as to the slope of theinteraction curves maybe extended to clamped pa
22、nels, because clampedpanels of appreciable curvature buckle at stresses equal to or onlyslightly higher than simply supported panels. (See figs. 4 to 7.)The interaction data computedfor simply supported panels aregiven in table 1.Empirical Interaction CurvesCurved rectangular panels are knownto buck
23、le in compression atstresses well below the theoretical values, whereas they buckle inshear at stresses close to the theoretical values; therefore, thetheoretical solution for the critical combinations of compression andProvided by IHSNot for ResaleNo reproduction or networking permitted without lic
24、ense from IHS-,-,-_CA TI_1928 5shear must be modified so that it may be used for the design of panelssubjected to combined shear and axial compression. Empirical interactioncurves for long plates with transverse curvature and for cylinders(references 3 and 4) indicate that the design curves in the c
25、ompressionrange for rectangular panels with substantial curvature would be of theform2(Rs)th + (Rx)e_ = 1 (1)where the denominators of the stress ratios (Rs)th and (Rx)ex p ar8_respectively, the theoretical critical stress of the panel in shearalone and the experin_ntally determined critical stress
26、of the panelin axial compression alone. Equation (i) should be conservative forall panels regardless of the length-_idth ratio and should become moreconservative as the ratio of th_ axial to the circumferential dimensionincreases.The critical shear stress to be used as the denominator of thestress r
27、atio (Rs)th may be obtained from figures 6 and 7. In orderto eliminate the need for interpolation for the critical shear stressof curved rectangular panels of any length-width ratio, the resultsof figure 6 for panels with simply supported edges are replottedin terms of other parameters in figure 8.
28、The ordinate in figure 8is the increase in the critical-shear-stress coefficient 2_ks overthe flat-plate value and the abscissa is a function of the curvatureparameter Z and the length-width rati_ of the panel _. In usingthis figure _ should be taken as equal to 1 whenever the circumfer-ential dimen
29、sion is equal to or greater than the axial dlmsnsion. Thevalue of the shear-stress coefficient for a panel is determined byadding the value of Aks found from figure 8 to the flat-plate valuegiven approximately by the equation4ks = 5-35 +-was obtained from reference 5 or more accurately by4ks = 5.34
30、+ -/B7“-“_Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 _CA TN 1928where _ is a/b. In a similar way a curve may be obtained for panelswith clamped edges by replotting the results of figure 7 in terms ofthe sameparameters as figure 8.The critical
31、compressive stress to be used as the denominator ofthe stress ratio (RX)exp maybe approximated by the design curves offigures 4 and 5 for cylinders and long curved plates, respectively.At very low values of Z (Z _ I0), panels buckle in compressionat a stress close to the theoretical value and the _h
32、eoretical inter-action curves may be used for design.Rectangular curved panels subjected to combined shear and tensionmaybe expected to buckle at stresses that agree closely with thetheoretically predicted values because tension tends to minimizeinitial imperfections. The theoretical interaction cur
33、ve shouldtherefore be used for this range.In connection with the present paper a set of 25 panels that hadbeen previously buckled were subjected to a combination of shear andcompression. (The previous results were presented in reference 6.)Becausemost or perhaps all of these panels had large initial
34、 eccen-tricities an inordinate amount of scatter in the various test resultswas found. Comparisonscould be made, however, between the differentcritical combinations of stress on each panel. These comparisons areshownin stress-ratio form in figure 9, in which the stress ratiosare based on the experim
35、ental stress for buckling under either compres-sion or shear alone. These comparisons confirm the shape of the curverepresented by equation (1) in the compression range.C0NCLUDINGREMARKSThe theoretical solution for the buckling of rectangular curvedpanels in combined shear and direct axial stress in
36、dicates that thebehavior of a panel is similar to that of a cylinder when the curvatureparameter is sufficiently high, irrespective of the length-width ratioof the panel. For lower values of the curvature parameter theoreticalinteraction curves for panels of five different length-width ratios arepre
37、sented that give either the shear or direct axial stress required forbuckling whena given amount of the other stress is present.IProvided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS -,-,-NACA TN 1928 7Because a panel having substantial curvature buckles in c
38、ompressionalone at a stress well below the theoretical critical stress, the theo-retical results for the critical combinations of stress are modified inthe compression range for the purposes of design.Langley Aeronautical LaboratoryNational Advisory Committee for AeronauticsLangley Air Force Base, V
39、a., June 23, 1949Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACATN 1928APPENDIXTHEORETICALSOLLVfIONThe problem of the buckling of a simply supported rectangularcurved panel under combined shear and direct axial stress (fig. l) issolved in a ma
40、nner similar to the one used for the buckling of a panelunder shear alone. (See the appendix of reference 1.)The equation of equilibrium of reference i is modified to introducedirect axial stress and becomes_4 w + Et _7-4 84w + 2vt 82w + _xt 82w = Or2 _x4 _x_y _x2where x and y are, respectively, the
41、 axial and circt:_ferentlalcoordinates.The problem is solved by use of the Galerkin method as outlinedin references 7 and 8. As in the case of shear alone, the followingseries expansion is used for ww = _ _ amn sin m_-_xsin n_ya bm=l n=l(A2)which imposes the boundary conditions of simple support.Div
42、ision of equation (AI) by D gives+ 84wCw 12Z2 V-4 -+7- 2ks _2 82w kx _2 82w+b2 8x_y b2 8x2=0The equation of equilibrium may be represented byQw=0Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 1928 9where Q is the operator defined byq=+_12z 2v
43、 -4m84 +2k8 _2 82 +kx _282b4 8x_ b2 8xSy t2 8x2According to the Galerkln method the coefficients are chosen to satisfythe equationssln p_xsln q-Qw dx dy = 0a(A3)(p = l, 2, .; q = l, 2, . . .)When the expressions for Q and w are substituted in equation (A3)and the indicated operations are performed,
44、the following set ofalgebraic equations results:_p2 + q2j32) 2 + 1_2 Z264p 4apq L_:4 (p2 + q2p2) 2 - kx_2p I32p3ks Z _- amn+ 7 m=l n=lmnpq(m 2 _ p2)(q2 _ n2)=0(p = i, 2, . .; q = i, 2, . .)where the summation includes only those values of m and n forwhich m p and n q are odd. The condition for a non
45、vanlshingsolution of these equations is the vanishing of the determinant of thecoefficients of the unknown values of apq. Thls infinite determinantmay be factored into two infinite subdetermlnants, one for p q evenand the other for p q odd. The vanishing of these subdetermlnantsleads to determlnanta
46、l equations similar to equations (9) and (lO) ofreference l, except that Mpq is now defined byProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-i0 NACATN 1928_2 _ 2 12 z2P424Mpq= _ p2+ q2_2) +321B3ks 7 (p2 + q2j32) 2 - kx_2p21These determinants give th
47、e combinations of stresses that causebuckling of curved plates with various ratios of axial to circumferentialdimension. The solution to the determinant where p q is evencorresponds to a buckle pattern that is symmetrical about the centerof the panel, and the solution where p q is odd corresponds to
48、 abuckle pattern antisymmet#ical about the center of the panel.By use of a finite determinant including the rows and columnscorresponding to the most important terms (usually ten are sufficient)in the expansion for w (equation (A2), the two determinantal equationswere solved by a matrix iteration method (reference 9) for the lowestcombinations of kx and ks that satisfied the equations. The presentsolution was found by maintaining kx at an assumed constant value andsolving for the lowest value of ks that satisfies each
