1、REPORT NO. 809PRINCIPLES OF MOMENT DISTRIBUTION APPLIED TO STABILITY OF STRUCTURESCOMPOSED OF BARS OR PLATESBy EUGENE E. UNDQUIST,ELBRmGE Z. STOWHLL,and ET-AN H. SCHUETTIISUMMARYl%e pn”na”ple$ of the Cross method of moment distribution,uhich hare preciously been applied to the stability of structure
2、8composed of bars under am-al had, are applied to the tabilityof structures composed of long plates under lon”tudinal load.4 brief theoretical treatment of the mbject, a8 applied to 8truc-ture8 compo8ed of either bar8 or plate8, i8 included, togetherwith an iilu8tratice example for each of these two
3、 types of8hwcture. An appendix presents the derivation of the formula8for the rariow86ti$nes8e8 and carry-orer factors used in soltingprobern8 in the stability of 8hwcture8 composed of long plates.INTRODUCTIONThe usual procedures for cahxdating criticaI bucldingloads for the members of compIex struc
4、tures are often some-what invoIved and are not- easiIy reduced to a set of routinecakulations. llany practical engineers, as a consequence,do not attempt to crdculate critical buckhg loads.One approach to the soIution of problems in the stabilityof structural members tht is puwly engineering in clwm
5、ct wwd that lends itseIf to simplified calculations is provided byuse of the principles of the Cross method of moment dis-tribution (reference 1). The theory of moment distribution,originally devised as a rapid method of stress analysis,desaibes how the resistance to an external moment, appliedat an
6、y joint in a structure composed of bars, is distributedthroughout the structure in accordance with the resistanteof the various joints to rotation. Tl Ioad in bar (absolute valueP CT2-( (9fixity coefficient in coIumn forrmda 2=7;stiffness factor (r)F,PLATES effective pIate modulus for stresses beyon
7、d the elasticrangeK Poissons ratiox half wave length of buckles in longitudinal directionb width of pIatet thickness of plateD flexural stiffriessof plate per unit Iength (MZ9) effective flexural stifkm of plate for stresses beyondthe ektic range (12:5,)57Provided by IHSNot for ResaleNo reproduction
8、 or networking permitted without license from IHS-,-,-58 REPORT NO. 80 9NATIONAL ADVISORY (20WTTEE FOR AERONAUT1(Xa longitudinal compressive stress in platek=u (always positive)MM*ewicreFWbending momentamplitude of sinusoidally distributed momentrestraint coefficientdeflection normal to plane of pla
9、teSUBSCRIPTSinitiaI valuecriticaleffectiveflangewebDEFINITIONSMember.The word “member” is used in this report toindicate either a bar or an infinitely long, flat, rect.anguhwplate.Jointi-A joint in a structure composed of plates, byanalogy to a joint in a structure of bars, is defined as theentire l
10、ength of the intersection line between two or morejoined platesStiffness and oarry-over faotortIf a bar is on unyieldingsupports at each end, the moment at one end negessary toproduce a rotation of one-fourth radian at that end is calIedthe stielI Oondltkm at farend orLdgebars.Iyv=-l Farcndorudge su
11、pported and srrbjcctcd to momentaqnaiand oppodtr to that mpllcdatmar end or c.The cpmtitiea S, CX,W, C11of this paper correspond8, C, f?, C“, respectively, of rcferenco 3.The stiflncss of abar computed according to the clrfinitionused herein is one-fouth that computed according h thedefinition used
12、by Cross (reference 1). In moment distrilm-tion the relative, not the absolutr, values of stifhwsscs ofthe members are of importance. The forrgoing definitionwas selected so that the stiffness of u bnr of constant crowsection with no asiid load nnd fiml at llJU fur cnd would IIc/L instmd of 41/L.Sig
13、n convention.-A clockwise moment acting on lhc endof a bar or at any station rdong tlw side cclgc of a plate ispositive and causes positlivc rotation at that und or station.An external moment applied at. a joint is considorcd to acton the joint.; a counterclockwise moment acting on a jointis positiv
14、e.CRITERION FOR STABILITYIt is assumed that all mcmbws in n atructurc compoacd oflmrs lic in the plane in which buckling occurs rind that hcjoints of the st.ructurc arc held rigidly in spare but arc frcwto rotate subjmt to the ekwtic restraint of tho connectingmembers. Simihwly, in a structuro compo
15、sed of plntcs it isnssume.dthtit the joints betwccu plntcs, or betwwm plntcsand longitudinal restraining mrmbcm, remain in thriroriginal ssraight lines but arc free to rotntc subject to theelastic restraint of the connecting rncmhws.In the discussion thut follows, either of two criterions forstabili
16、ty may be used. For cnch criterion, the stifliwss andcm-t-y-overfactor nrb functions of the nxial Iord in tlw lmror the longitudinal load in the phito. (See rrfwrnces 2, 3,4, and 5.)StifFnessoriterion for stability.-From a structure of manymembers the section comprising mm joint shown in figure 1is
17、considered. Figure 1 may bu intwpro W m being eithera plan view of a structure composed of btirs or nn cnd viewof a“structum composed of long plntes. AI cxternnl momentof 1 is assumed to be applied at the joint i. If thv SlUC-turc is composed of platea, tltis moment is the extermdmoment per unit Ien
18、gth at the station under considcrntion.Because4he angles between members at the joint nro wc-served and the rotations of all members at the joint musttherefore be equal, the moment of 1 nddul to Imhtnco thisjoint is distributed among the members in proportion to theirstiffnesse9, as follows:flu* to
19、member ijlXYo (1)The condition of neutral stability gives the critical bucklingIoad for the structure and is obtained by setting.the stiffnessstabty factor equal to zero, orSY,=o (2)lrI the genertd case there is more than one criticaI bucklingIoad; thus, satisfaction of equation (2) is hsticient for
20、 thesolution of a given stability probkm. Instead, the IovwstIoad that satisfies equation (2) must be calculated andcompared with the Ioad for which the structure is designed.Only if this lowest criticaI Imd is greater than the design“load is the structure stable.W“-”%).-J3FIGCUIEI.+?ectkn comprisin
21、g one Jotnt.According to the definition of stitlness, the moment dis-tributed to any member must be the rotation of the jointrmdtiphed by the stiffness of the member. Hence 0, therotation expressed in quarter-radians of joint i caused bythe extermd moment 1, is(3)Equation (3) will be used under the
22、section “31ethod ofJfnking Preliminw Estimate of the Criticrd Load.”Series criterion for stabiIity.-In a structure of manymembers, the section comprising two joints shown in figure 2is considered. An externaI moment of 1 is assumed tobe applied at joint f. If the structure is composed of plates,this
23、 moment is the external moment per unit length at thestation under consideration. By a momentdist.ributionanaIyeis of reference 3, the totaI moment in members E=p . -11724. For each load in cnch of the members, dctcrrninc tlwvalue of the terms required to evaluate cquation (19), by useof the tables
24、of reference 3 or 4.5. The assumed load that gives r= 1 is the criticnl burklillgload.The results of this procedure M applied to the proldcm offigure 5 me given in table I; the values of c in the first columnare given for reference only and, as stated in paragraph 1 ofthe foregoing procedure, were s
25、o assumtitlthat. n series ofreasonable values for the ax;al loml P in the compressionmember bc co.uId be obtained. In the last column of ttiJhJ1fire given the valurs of r correspond ng to the nssumcdvalues of c. As the valuc of c inertascsfrom 1A to 2.6, the,value of r increases from 0.133 to 1.63.
26、If the dnta of table Iare plotted, it is found that when r= 1 the Iowcst critimlbuckling loads for the trial design tirem, bc, and de- 10,260 compressionab and cd- 8,890 tensionThese critical loads arc grmtcr than the loads to which therespective members are subjected (see fig. 5). The tubeselected
27、for the trial design is thcrefom stable and thr marginof safety for tlm system is10260 ;_8890.9940 l=o.03A sgle margin of safety is obtained for the whole systwnregardlessof which member is used for its calculation bccausc,when the critiral 10M1is reached, all members deflect.More than one type of i
28、nstability is possible, thcorcticaly;therefore, as the loads F increase, there is more thrin onevalue “ofP for which r=l. (See table I.) For rach type ofinstability there is a corresponding critical loud. In Imign,however, the lowest critiml load should bc cahmhitcd andcompared with the loads given
29、in the problcm.Tal.deI shows further that, for values of c bet.wcen 1A and1.5, the value of S1,changcn from positivc to ncgal ivc.According to the stiffncss criterion for stability, this chtinguof sign means that members de and ej, considered alonoj havochanged from stable to unstable, It is also no
30、ted thnt SIchanges from positive to negative for values of c between 2.6and 2.7; members cd, de, and ej, considered alone, htive there-fore changed from stable to unstable, hut at a much higherload. The change from stable to unstable for all membersoccurs for values of c between 2.5 and 2.6 when r=
31、1.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-PRINCIPLES OF MO.MENT DISTRIBUTION APPLIED TO STABILITY OF STRUCTURES CU31POSED OF RARS OR PLATES 63Structure composed of pIates,The critical compressivestress for local instabdity of a 24ST ahunimm-d
32、loyZ-section COIIUUIIwith the cross-sectional dimensions shownin figure 6 is to be determined.It is convenient in symmetrical plate problems of thistype to use the modification of the stiffness criterion forstabdity previously discussed. If opposing unit extermdmoments are applied at the joints betw
33、een the web andthe flanges, the stifhmss stabdity criterion, as gien byequation (9), iszFf,=WI.+Fvw=o (20)where the subscripts F and IT refer to the flange and theweb, respectively.cb=l 7 tw.o.05FIGCEX6.IUustmtlve plate probkm.The tables of reference 5 give the values of S1l and Win the dimensionles
34、s form SII/(/b) and fTv/(/fJ) ratherthan directly. It is therefore desirable to write equation(20) in the formFi=%?+itmw=o(9If this equation is divided by , it becomes$=u=t%e)(z)”Because W/(tw/t.)a, the stabihty criterion may bewritten in terms of the mod the sign convention used, asdistinguished fr
35、om that given in the section on “Definitions,”corresponds to that of reference 12, in vrhich deflections w arepositive downward and a moment is positive if it producescompression in the upper fibers.General deflection surface of a plate buckled under com-pression.Before the values of stitlness and c
36、arry-over factorfor flat plates under -rarious conditions of edge restraint maybe computed, the deflection surface of a flat plate buckledunder a compressive load with a moment applied along oneunloaded edge must be described.An infinitely long flat plate under ongitudimd comprtionis shown in thesck
37、tion of equation (Al) is therefore taken in the formThe unknown function (g) may be determined by sub-stituting the expression for w into equation (Al). It isfound that the function must satisfy the equation(A3)Equation (A3) is an ordinary diflerentid equation of thefourth order, the solution of whi
38、ch isj=C, COSh+h Sinh +ti COS+C, Si.U (A4)where cl, G, Cz, and CLme arbitrary constants and“”AI=.FIGGEE S.-InSnitely long ftat te under longitudinal cempmsskt.The deflection surface of the plate is riow found byb-st.ituting this result for f in equation (A2):In this solution, feur conditions may be
39、imposed along theunloaded edges to fix the four constants. One of the fourconditions wiIl ahvays specify the presence of a momentiLIOCOS along the near edge, and another will speeify titthe deflection w along this edge is zero. The remaining twoconditions wiIl be m.ried to suit the conditions at the
40、 faredge of the plate of which the stiffness is bejng computed.05Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-66 REPORT NO. 80 9NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSStiffness of a plate with far edge fixed.-Figure 9 showsa flat rectangular p
41、late under compression with a moment Mapplied along one edge at y= $ and with complete restraintagainst rotation along the paralleledge at y=” The stiflnessSof the plate is defined as_ M()6 (A6)v.+where (6) is the rotation of the edge at y expressed9-+in-quarter-radians.The general expression (A5) f
42、or th deflection of the platemust be specialized to the case of figure 9 in which theboundary conditions are:(V-*; =0 (A7)D($+lgg =M=MfJ Cosy (A8)u-+()%U.;=o (A9)FIGE 9.-P1Etewith momentaIIPHodatIImrmke,fmcdreW.After determination of the arbitrary constants in equa-tion (A6) by use of these boundary
43、 conditions, tho defhctionsurface for the case of figure 9 is found to beLFrom this deflection surface there is obtained1(0_;=4 ()y._$=* D(dT)1 + 1a:tanh + tan cccoth (fig, 9) to the moment at theand the result substituted in equation (A 13), it is found thatProvided by IHSNot for ResaleNo reproduct
44、ion or networking permitted without license from IHS-,-,-PRINCIPLES OF MOMlmY17 DISTRIBUTION APPLIED TO STABILITY OF STRUCTURES COMPOSED OF BARS OR PLATES 67_The moment at the near edge is, from equation (A8),(A15)By definition, the carry-ova factor is(A16)with the si=m of the moment at the far edge
45、 changed toconform to the sign convention given in the section“Definitions.”*rFIGLTiE 10.Plate with moment appl!ed at near edge, far edSCIMnge: b,- (A17)where (8) b is the rotation of the edge y= espresaed inr-yquarter-radians. The general expression (A5) will again beused to compute 6 and the bound
46、ary conditions will be:(W),=*$=O (A18) VW )TX +P g =M=lifi.1 Cos .(A19)1-;20)By use of these boundary conditions, the arbitrary con-stants in equation (A5) may be computed, and it is foundthat the deflection surface for the case of re 10 isFrom this deflection surface, the magnitude of the rota-tion
47、 along the edge y= is found to be()(6),.-:=4 ,=-gZM-b =D(cF+) ( a tanh + B tan +a coth )/3 cot $(A22)where 8 is expressed in quarter-ratilans. upon substitutionof this expression for o in equation (il17), it is found thatAccording to the boundary condition given in equation(A20), there is no moment
48、at the edge y=; Hence, thecarry-ov= factor IW with the far edge hinged is zero. Stiffness of a plate with far edge free.-Figure 11 showsa flat rectangular plate under compression with one edgeV=ZI free and a moment .liapplied to the paraI1el edgeg=O. The stiffness of the p!ate is defined as(A24)wher
49、e (19)U.0is the rotation of the plate along the edge y=Oand is expressed in quarter-radians.The general e.xpr-ion (A5) is used to compute therotation 6. The boundary conditions for a plate with faredge free are:(W),.,=o (A25)(A27)Provided by IHSNot for ResaleNo reproduction or networking permitted withou
