1、- 7 . , : .J ;-, 2 - , I- : .- _,. - -, . _. -. / ,/. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.I ./ / . . ., -, ;t “ _ - , , ; ,;“ “; , Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT 9
2、34 RECOMMENDATIONS FOR NUMERICAL SOLUTION OF REINFORCED-PANEL AND FUSELAGE-RING PROBLEMS By N. J. HOFF and PAUL A. LIBBY Polytechnic Institute of Brooklyn Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-National Advisory Committee for Aeronautics Hea
3、dquarters, I%J F Street NW., Washington. 95, D. C. Created by act of Congress approved March 3, 1915, for the supervision and direction of the scientific .stndy of the problems of flight (U. S. Code, title 50, sec. 151). Its membership was increased from 12 to 15 by act approved March 2,1929, and to
4、 17 by act approved May 2.5, 1948. The members are appointed by the President, and serve as such without compensation. JEI growing- unit method is rccommcndcd. The lst.ter can be applied only to panels the boundary conditions of which arc spccificcl in terms of forcr at least at one Cal of the strin
5、gers. Table 6 is the rclasatiou table in which thrsc group oprra- tions arc usccl. The group operations given in table 5 require some es- planation. In order to avoicl introducing a J7B-rcsiclua1 when joint A is rclasccl by application of operation (l), a v,-displacement is applied, the magnitude of
6、 which can be calculated from the equation The growing-unit method applird to reinforcrcl panels is as follows: Thf joint at. the free Cal of an arbitrarily sclcrtcd unbalancrd stringrr, callctl hcrcinafter thcl princGpa1 joint and the principal stringrr, respect ivcly, is tlisplacctl so as t 0 liqu
7、idate the rcsiclual on this joint. At the samcl time the joints lying on acljacrnt parallrl stringers and tlir same transvrrsc stiffener arc clisplacctl so that the rc4clun.ls that would be otherwise introduced by shrar from the balancing of the principal joint as well as any external forces applied
8、 to thcsc joints are likewise liquidated. In the second opera- tion the nest joint on the principal stringer is rc4axecl while the previously balanced joints on thr first transverse stiffenrl and the joints on the second transverse stiffcncr arc kept in balance by suitablr displacements. After this
9、scconcl opera- tion no residuals remain at the joints of the first two trans- verse stiffeners. After a sufficient number of repetitions of the procedure all residuals will be confined to reaction points or will bc liquidated; the panel will then be in equilibrium. -55.2,u,+2.00=0 (2) Thus operation
10、 (9) is =(2/55.2)=0.0362 and (10) is a group operation equal to the sum of operations (1) and (9), which liquiclatcs the rcsiclual Y, without introducing a YB-unbalance. After operation (10) is used, unbalances exist at joints E ancl F, that is, on the second transverse stiffener. In order t.o balan
11、cr thcsc without disturbing the recently cstablishccl balance at A and B, two group operations are clevclopcd: One permitting the balancing of E and one per- mitting the balancing of F. The magnitudes of v, and vg required to maintain the balance of A and B when a displace- ment of v,=l is undertake
12、n arc given by the following equations: -50.8v,+2.00v,+46.8=0 (3) 2.OOvn-55.2v,-+-2.00=0 This procedure is demonstrated on the panel shown in figure 3. The physical properties of the panel are the same as thosr of the previously discussed panels rsccpt for the additional bay in the direction of the
13、axial forces. Actually the convergence of the relaxation method for this panel would be quite rapid, but for convenience the growing-unit method, applicable when this convcrgcnce is slow, is demon- strated thereon. Table 5 is the operations table for this panel and contains not only the individual o
14、perations but also the group operations of the gsowing-unit method. 1, II I I I III I III II II II411 1111111111 -.-. , _ _. - .- These arc satisfied by v,=O.921, operation (ll), and v,= 0.0695, operation (12). Operation (13) is therefore estab- lishecl as the sum of operations (3), (1 l), ancl (12)
15、. The magnitudes of v, ancl vB required to maintain the balance of A and B when a displacement of v,=l is undertaken are given by the following equations: -50.8v,+2.00v,+2.00=0 (4) 2.OOva-55.2v,+51.2=0 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
16、6 REPORT 934-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS These are satisfied by v,=O.O758, operation (14), and v,= 0.923, operation (15). Operation (16) is the sum of opera- tions (4), (l4), and (15). Since group operations (13) and (16) both introduce Y,- and Y,-forces, the magnitudes xl3 and X, of
17、 these groups required to liquidate the - 11 l-pound and -Q-pound residuals at E and F, respectively, are given by the following equations: -58.3r+9.4-111=0 I (5) 9.4x13-62.6x,-Q=O Thus r13= -1.975 and x,= -0.444. Joints E and F a.rc balanced without disturbing the balance of A and B by the use of t
18、hese multiples of operations (13) and (16). In eliminating t,he residuals at joints J and K multiples of operations (13) and (16) are applied since these operations permit displacements of E and F to be undertaken while the balance at A and 6 is left undisturbed. When joint J is displacrd a unit amo
19、unt, multiples of operations (13) and (16), defined by the following equations, are used so that the balance a.t A, B, E, and F is maintained: -58.3+9.4+46.8=0 1 (6) 9.4x,3-62.6x,6+2.OO=O The solution to thcsc equations is 2,3=0.828, operation (17), and ,=0.158, operation (18). Operation (19) is the
20、 sum of operations (5), (17), and (18). In a similar manner all the individual and group displace- ments described in table 4 are found. It may be mentioned that in the present example no shearing stresses were set up in the middle bays because of the symmetry of structure and loading. The original
21、operat,ions table was already established in a manner which complied with these rcquire- ments of symmetry. When such is not the case or when there is a greater number of stringers in the panel, displacc- ments of principal stringer joints will, in general, cause residuals to appear at more joints s
22、o that three or more, rather than two, simultaneous equations have to be solved at each step. NILES TABLES In reference 10, Kilts demonstrates for the solution of rrin- forced-panel problems a method which essentially parallels the previously described relaxation method. The Xilcs method is a proced
23、ure for balancing a stringer by the usr of tables which give the displacements of each joint on the stringer required to liquidate a residual on a given joint of the stringer. The tables are worked out for various end conditions and sheet shearing rigidities. Since reference 10 contains tables only
24、for sheet of rela- tively low shearing rigidity, the Niles method is limited in this respect in the same way as the relaxation method. However, the tables can be employed on stringers with the boundary conditions at both ends specified in terms of dis- placement; for such problems no step-by-step ro
25、utine relaxa- tion method has beer1 recommrnclrrl. Also by use of the tables exact balance of a stringer is gained after a single displacement of each joint, whereas in the relaxation method, because of the shear, small unbalances remain after each joint is moved. On the other hand, t,he relaxation
26、method can be applied to stringers with irregularly spaced joints for which no tables were set up by Niles. Since in reference 10 several examples of the procedure are given, no application of the Niles method is shown herein. ELECTRIC ANALOGUE Another convenient method of solving the problem of for
27、ce distribution in a reinforced panel is that in which the voltages are measured in an electric network which is so constructed as to make it a complete analogue of the reinforced panel. When suitable electric equipment is available, an analogous uetwork can be hooked up and testecl with very little
28、 work. A particularly attractive property of the stress-analysis procedure by means of electric measurement is the ease with which the effect upon the stress distribution of changes in loading and in dimensions of the various structural elements of the reinforced panel can be investigated. This perm
29、its the development of an efficient design with little analytic work. The analogy between the forces transmitted through the different structural elements of the reinforced panel and the currents flowing through the various branches of the direct- current network can be explained with the aid of fig
30、ures 4 and 5. The problem investigated is the so-called “one- dimensional shear lag.” It is assumed that the transverse stiffeners are infinitely rigid so that the vertical, or longi- tudinal, displacements v alone need to be determined. The portion of the sheet covering considered effective in tens
31、ion or compression is added to the cross-sectional area of each stringer and the panels of sheet are assumed to carry shear stresses only. A consequence of these assumptions is that the shearing stress must bc constant in each panel. Y I i 2 3 4 5 I 6 t b I -7 3 Prr,nm I.-Fo:ces transmitted through
32、structural elements of reinforced panel. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-, _,!._ ,-Y .l; , RECOMMENDATIONS FOR NUMERICAL SOLUTION OF REINFORCED-PANELS AND FUSELAGE-RING PROBLEIviS 7 The analogous direct-current network contains as man
33、y binding posts as the number of joints in the reinforced pantll. Adjacent binding posts arc connected by conductors having prescribed resistances R. Predctcrmillctl clcctric currents I, which correspond to the forces E applied to joints A and 6 of tlic reinforced panel, are introduced into the nctw
34、orlr at points A and B. It is now rrcallctl that in thr relaxation method the joints of tlit panel are first assumctl to 1) rigitlly fiscal to a rigid wall bchintl the pancxl. Thcl cxtcrual loads arc first applictl to these rigid pegs, rcfcrrctl to as the “constraints.” The panel is obviously in equ
35、ilibrium under thcsc conditions but this artificial equilibrium is entirely different from that prc- vailing in the actual panel, which is not attached to any rigid wall. The actual state of equilibrium is approached by the step-by-step procedure of the relaxation method, in each step of which one s
36、ingle constraint is removed and the cor- responding joint is displaced until it reaches its equilibrium position in the system in which all the other joints are still rigidly fixed. For instance when joint 1 of the reinforced panel is moved through a distance v in the positive direction, this displa
37、ce- ment imposes forces upon all the adjacent joints numbered from 2 to 9. Three typical forces are givrn by the equations: 863931-50-Z F =v?bt 91 4a (S) F zvGbt 61 2a (9) whcro Fsl, Fgl, FBI forces acting upon joints 8, 9, and 6, respective- ly, because of displacement of joint 1 E modulus of elast
38、icity of stringer G shear modulus of sheet t thickness of sheet V displacement of joint 1 In the case of the analogous network it can be assumed that the potential of each binding post is zero at the outset. If there is no potential difference, no current flows between the posts. It can be imagined
39、that the current,s introduced at points A and B are taken out of the system by means of some imaginary conductors. However, the actual distribu- tion of .currents in the network prevails without the aid of the imaginary conductors. This actual state can be approached also by means of a step-by-step,
40、 approximation- type calculation. For instance it can be assumed first that the potential of binding post 1 is elevated to the value V. After this change there is a potential difference between binding posts 1 and 8 and consequently a current will flow from post 1 to post 8. The magnitude of this cu
41、rrent can be calculatctl from t.hc equation I,= Jr/R,= P8,1 (10) where R, is the rrsistance and CY8, = 1/R8, is the conductance of the conductor between posts 1 and 8. Similarly the cur- rnit flowing from post 1 to post 9 is Ig1= cg, v The current Rowing from post 1 to post 6 is (11) I6l=c,v 02) Com
42、parison of equations (7) to (9) with rquations (10) to (I 2) reveals an analogy between the effects of a displacement 2) of joint 1 and thr raising of the voltage of binding post 1 by an amount V. The current caused by the change in potential corresponds to the force caused by the displace- mcnt, pr
43、ovided that the conductance of each conductor is made equal to the influcncc coefficient in the corresponding force equation. Hence c =G 61 2a (15) In the relaxation procedure the equilibrium state is approached by displacing individually the joints and Tsum- ming the effects of each displacement. I
44、n exactly the same way the actual distribution of the currents in the network can be determined by changing individually the voltages of each binding post and summing the effects of these changes. In the reinforced panel equilibrium is obtained when at each joint the sum of the external forces and o
45、f all the internal Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 REPORT 934-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS forces caused by the displacements is zero. The forces arc considered positive if they are directed as the positive dis- place
46、ments. In the form of an equation, zF=o 06: An analogous equation in the direct-current network ir: furnished by Kirchhoffs first law, according to which the sum of the currents flowing into any binding post must be zero. Currents in the direction of any binding post are considered as positive. In t
47、he form of an equation, 2I=O (17) Comparison of the last two equations reveals that the conditions of equilibrium for the reinforced panel and Kirch- hoffs first law in the case of the direct-current network complete the analogy of the two systems considered. It is possible therefore to construct an
48、 electric network with the same configuration of binding posts as that of the joints of the reinforced panel. The conductances of the conductors connecting the binding posts must be so chosen as to make them proportional to the corresponding influence coefficients in the operations table of the rein
49、forced panel. If then cur- rents are introduced at the binding posts which correspond to the joints at which external loads are applied, the distri- bution of the currents in the network will be the same as the distribution of the forces between the various structural elements of the reinforced panel. In the first applications of the relaxation process to
