1、,. 86ARRIVO. 3K04,. .,NATIONAL ADVISORY COMMITTEE-.FOR- AERONAUTICSWIAIHIMEImwmORIGINALLYISSUED?Joveniber1943 asAdvance Reetrlcted Report 31kCEARTSFoR CM.cxnAmm OFTEE CRITIOALSTRESSmRIOCAL INSTABIKCl!YOF COIZJMXSWITH I-, z-,By W. D.CHANNEL,JU!lDRECTANXJIAR-TUBESECTION=.Kroll, Gordon P. Fisher, and G
2、eorge J.Iaagley Memrfal Aeronautical IdxmatoqyIangley Field, Va. ! -,.,:Eelmerl .:”,-,.,. .- . -.,-. , eI ,./NACA WARTIME REPORTS arerprintsofpapersoriginallyissuedtoproviderapiddistributionofadvanceresearchresultstoanauthorizedgrouprequiringthem.forthewar effort.,Theywerepre-,viouslyheldunderasecur
3、itystatusbutarenowunclassified.Some ofthesereportswerenot”teeFi-nicallyedited.Allhavebeenreproducedwithoutchangeinordertoexpeditegeneraldistribution.L - 429-. . -., .:,.:.,. ,. :,., . _ ;Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-3 1176013543070
4、3U!J?IC)NALADVISORY COMMITTEEADVANCI! REShlICTtiDYOR AERONAUTICS._-.REP”oR”-CHTs FOR CALCULATION OF THE CRITICAL S!f!R?iSSFORLocAL 1NSTABILIT% or COLUMNS WITH h, zk,CHANNEL, AND RECTANGULAR-TUBE SEOTIONBy W, D6 Kroll, Gordon P. Fisher,SUMMARYCharts are presented for theical stress for local instabil
5、ityand George J. Heimerlcalculation of the crit-of columns with 1-. Z-.channel, and rectangular-tube section. These charts-are-intended to replace the less complete charts published inNACA Technical Note No. 743. !The values used in extend-ing the charts are computed by moment-distribution methodsth
6、at give somewhat more accurate values than the energymethod previously used and also make it possible to deter-mine theoretically which element of the cross section isprimarily responsible for instability.An experimental curve is included for use in takinginto account the effect of stresses above th
7、e elasticrange on the modulus of elasticity of 24$-T aluminum alloy.A determination of the dimensions of a thin-metalcolumn for maximum critical stress with certain given oon-ditions is presented.INTRODUCTION,One of the important requirements in the design ofthin-metal columns for aircraft is the de
8、termination ofthe critical compressive stress at which locai instabilityoccurs. Local instability of a column is defined as anytype of instability in which the cross sections are dis-torted in their own planes but are not translated or ro-tated.Provided by IHSNot for ResaleNo reproduction or network
9、ing permitted without license from IHS-,-,-2The critical stress for local instability canusually be given in terms of the geometry of the sec-tion, the properties of the material, and a coefficient,Reference 1 presented charts for the determination ofsuch coefficients for columns of I-? Z-$ channel,
10、 andrectangular-tube sections. These charts, however, con-tained relatively few curves and in some cases requiredinterpolation over a wide range.In order to make the charts of reference 1 morenearly complete and to reduce the necessary range ofinterpolation, each chart has been extended to includeei
11、ght intermdiate curvese The values used in extend-ing the charts are computed by moment-distributionmethods that give somewhat more accurate values than theenergy method previously used and also make it possibleto determine theoretically which element of the crosssection is primarily responsible for
12、 instability.The present report includes the extended chrts,aldng with tables of the values used in preparing thecharts and is intended to supersede refereqqe 1. Anexperimental curve is included for use in taking intoaccount the effect of stre=ses a%ove the elastic rangeon the modulus of elasticity
13、of 24S-T aluminum alloy6A determination of the dimensions of a thin-metal columnfor maximum critical stress with certain given conditionsisAb5Ezh.presented,SYMBOLScross-sectional area .width of end or narrower wall of rectangular tube orof plate element of 1-, Z-, or channel sectioneffective flexura
14、l stiffness of plate per unit length-.r3r)Iztt12(l - Ma)“ 1modulus of elasticitywidth of side or wider wall of rectangular tubeProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- . 3k.k sectS111AMcrmnondimensional coefficient dependent upon relativedime
15、nsions of cross sectionsection coefficientthicknessstiffness in moment-distribution analysis for faredge free (no support and no restraintagainstrotation)stiffness in moment-distribution analysis for ffaredge supported and subjected to sinusoidallydistributed moment equal and opposite to momentappli
16、ed at near edgerestraint coefficient, a measure of relative resis-tance to rotation of restraining element at edgeof platehalf wave length of bucklePoisson!s ratiocritical compressive stressnondimensional coefficient that takes into accountreduction of modulus of elasticity for stressesabove the ela
17、stic range. Within the elasticrange, n = 1.Subscripts:J? flangew webb end or narrower wall of rectangular tubeh side.or wider wall of rectangular tubeFORMULAS FOR CRITICAIj STIKESSFor an I-, 2-, or channel section, either of twoformulas given in reference l,may be used for calculatingthe critical co
18、mpressive stress. The two formulas are,., _.-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4andacr = k2Et F2- -.-11 12(1 - a)ba(1)(2)The corresponding formula for a rectangulartube sectionis given in reference 1 asIn using formulas (l), (2), andare
19、 shove the elastic ranges gcr/11-.(3)(3) when the stressesis first evaluated,and cr is determined from this value by means of thecurve of figure 6. The relationship between cr andUcr/TI will be further discussed in another section ofthis report.DISCUSSION OF CHARTSAll of the quantities on the right-
20、hand side,ofequation (1), (2), or (3) are known except the value ofthe coefficient kw, kF$ or k. This value may be readfrom the appropriate chart (figs. 1 to 5) after the nec-essary dimension ratios are computed and applies wheneverthe length of the column is greater than several (3 or 4)times the w
21、idth of the widest plate element:,In general, when a column of 1-, 2-, channel, orrectangular-tube section fails by local instability oneof the two elements (web and flange or end wall and sidewall) of the cross section may be said to be primarilyresponsible for the instability; that is, as the load
22、approaches its critical value, this one element is noProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-5!)J .,longer capable in itself of supporting the loads imposed,L3 oh i% $ithout%ucklifig atidreqti”iresa-cetitain-”iitioizntofi,1 restraini from the
23、 other element of the cross section inorder to delay buckling until that load for which the:1;.p; CI?OSS section as a whole becomes untale is reached.j The charts show which element Of the cross sectfon is be-iiing restrained against buckling by the other element. Adashed line is drawn on each of th
24、e charts (figs. 1 to 5)r connecting the points” for which the two eler,ents are+I ./ equally responsible for the instability Of the section.,; This line divides the chart into two regions: In one,.region the web (or side wall) is primarily responsiblefor instability and in the other region the flang
25、e (orend wall) is priharily responsible for instability. AI column with a given cross section will fall into one ofthese two regions, depending on the values of the variousdimension ratios.RI!JLATIONSHIP BETWEEI? UCr AND OCr/q. .Yigure 6 shows the relationship between cr and/cr T asdetermined from t
26、ests of 243-Taluminum-alloycolumns of Z-, H, and channel section, either formed fromflat sheet or extruded. This figure was prepared by plot-ting the experimentally determined values ofCcr as/ordinates against the values ofi/acr Q as abscissas. Thevalues of ucr/q were computed according to equation
27、(1),Jand the chart of figure. 1:or. 3. The results of the tests.,.,!are di-scussed in more detail in reference 2,iiSimilar expeimental data for materials other than24; (see, reference 4), wheremr=% =-”12(1- W2)The formula isX48. With the values of c from step 7, determineF from the chart of figure 3
28、, reference 4*9: Flot F from step $ and kF from step % as , ;“ordinate against eithe,r of the two. values as ab”scissacThe intersection of the two curves gives the correctvalue of kF for the particular value of A/b.2.D. Repeat “steps 2 to 9, assuming different valuesof /bW., . ,11. Plot the values o
29、f kF from step !3againstA/bp. The minimum of this curve, gives the required valueof k.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-If the calculations indicate that S%vw” is nega-tive$ the prediction” that the flange is the primarycause of instabi
30、lity is wrong, In such a case, thecalculation must be carried out with S111 F instead ofs=J .in step 6, and with the chart of figure 3,.refer-ence 5, in step a. In addition, all the subscripts Ywill become W, and vice versa.The results of the procedure outlined herein asapplied to the problem of fig
31、ure 7 are given in table 1,The values of k, in the last column of table I weredetermined according to step 9. If these values of kare plotted against A/by, the iiiinimumvalue is foundto be about 0.73. The value of kw can be computed fromthe formula given in step 5.Tables II to VI give the minimum va
32、lues of kws kF#and k used. in the preparation of figures 1 to 5. Allof the values of k and. kw in these tables except thosemarked a were computed. either by the method just out-lined or 3Y the momentdistribution method discussed inreference 3. The values of kF were then computed bythe eque,tion give
33、n in step 5* The values narked a arethose computed by the energy method and. used in the prep-aration of the charts of reference 1.DIMENSIONS OF TEIN-lViETAL COLUhNS I?ORMAXIiqUM CRITICAL STRESSEquation (1) gives the critical stress for an 1,z-, or channel column in terms of the width and the thick-
34、ness of the web. The effect of the presence of flanges istaken into e,ccount in the evaluation of the coefficientkv a71 For the purpose of studying the dimensions that givemaximum critical stress, the form of equation (1) is pre-served but the concept of certain terms is generalized,The ratio b/t of
35、 a plate may be called the aspectratio of the plate. A corresponding quantity that expresses the Ilsection aspect ration for a thinmetal col-umn is the area of the section divided by the squa”re ofsome thickness If, therefore, eq,uation (1) is writtenProvided by IHSNot for ResaleNo reproduction or n
36、etworking permitted without license from IHS-,-,-B,Ail.9.ucr = kec= Tr23i-v 12(1 “”v2)t,:p ()-” -A“(4) then the value of the section coefficient k.ec isa,1measure of the effect of the shape of the section b/b”W, on /UC= n for a given section aspect ratio A/tw2 andn,!1 a given value of *w/ty. In orde
37、r to show ”that ksec is,1; dependent otionly%?/% and tw/tF$ equation (1) isset equal to equation (4), with ,the result that(5)From the geometry of the section (Z or channel),A= bvtw + 2bTtE. If this value of A is substitutedin equation (5) and the equation is solved for ksec,the result is.:(6)! The
38、value of kW depends on only .bF/bW and tw/ty,i and the value of kec therefore also depends on only:y.h these two rtios.In figure 8 the values of kec as determined ,byequation (6) are plotted fo FOR I-S3CTIONSmT0.5 0.6,.1.288-a.623-a.567-.547?.59:5;g .56i.? .550.198Jm2: :;-O:A:$.590.568-.528-.?.y).36
39、6.271.107-.-.-0.685J#.605-.567-.;i:416;$;-.-“- z0.29:673.651-.611-:5;Ji.60:;:-.- -a.852- -1.021.00 1.982.985-al.1471a71120I- - -0.960a71957:8%.7770.883; E#g.703 .7981.01*Oz?1.041.Ch 1.119-1.096I- -!-.- -.762-.685.615:M-.653-.5732:?2.304.2ooa.112.8621.96211.04911o98- 1-”1-: I - - - - - - - - - - - -
40、-.799a71 371:42;-aComputed by energy solution (reference1).TABLE lV-C.LLCULATEDHT.NIMUMVALUES OF % FOR CHANNELWD !Z-SECTIOI:S1i/t.bF/bo.050.100:H;.179.192.200.227.250.3002%.450JL;:.525.550.560a71575.600.800.co1.000.34.00I- - - - a4.oo-a4.oo35.46-%&; -94.016.02- . -26.19- 34.oil-S6.31-a4.58- n,-a4.59
41、-X6. -a4.60-I I- - - -a6.1+3 %:gil:=-6.50 p; ;:j- .6.533:743.25- -6.5/50II- - - - -4.00-a3.98-a3.97-a3.28- a3.63.81 3.76.3.48 3.26b2.78 2.12.18 1.7II- -I- - i- - - -6.035.595.154.79:;:2:$ : $!?6:96 1.9311:193:70 I I I- - - - - - - -2.02 1.69- -1.19:0 ;:7.8cra71) .)A- -1.42 1.19-l Ii.U61 - l-l - l- i
42、- 1- 1-,- - ,-.-1- 1-1 - -,- 1- 1-1 - l-.- l- -= -.98 .83.72-.55 4:.36Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-IJACATABLEVCALCULATEDMINIMUMVALUESOF. kF FOR CHANNELAND Z-SECTIONSw“1.6 .8 2.0- al.28-0.8 0.9 t#h0.2k“.:81.01.250tI.29I671.759iI.861
43、.181.9052.0002.1052.2222.0032*57t:;:;4.40Q-4.8005.2005.6006.0000.6 II1.0 1*2 1.40.5 0.7i1.2881.111-a.962-al.288-a.695-=62:?Z;.545.513.502.i01:$;-.26-.146-a.083-a.059-a.out-3.650.617.592.559-.491jy:.351:;:.136-0.062.,76.655.611-.543-.489y;i.27.171-I- -.-1- - !%.23_-.-% .201.1341.161 1.191.110l.fio 1.
44、151.1101.143L.0991.137I-t- -L- -.-k3.7720.836.726 .791i .767:;? .727- -.8900.98.870 .96?.832a71934.792 .911-1.071.0!/t1.051.015 1-1.19-1.18-1.11.1z1.17%ae68;.5;a“.I- 1- 1- -L.084-;- 1- - 1- 1- -I-I I- - ._- - - -.72s .864 .990-.- -.598 .657 -1.147.-1- -A-;:;: a.236a.199a.170a.146a.127- I.- -.748.644.515.377.068:6;.489-1.1311.1041.013.772I- -
