Document
REPORT No. 92
STRESSES
ANALYSIS OF WING TRUSS
INCI.UDLVG
OF ItEDuNDmTcIEs
By Ii l?. WAIUNER and IL G. MILLER
Aerodymarnimi Laboratory, National Advisory Cotittee for Aeronautics, Langley Field, Ya.
REPORT NO. 92.
ANALYSIS OF R711NGTRUSS STRESSES, By EDWARD P. WARNER and ROY C+. MILLER.
A.erodynamical I,abratory,N. A. C. A., Langley Field, Va.
—— This repori -was prepared at the Langley NemoriaI Aeronautical Laboratory of the National Aclvisor,y Committee for Aeronautics under the direction of the f20mmittee on Aerodynamics by Edward P. Warner and Roy G. Miller.
It has been the usual practice of airplane designers in making structural analyses to treai the airplane, not as a collected whole, but as an assemblage of separate units, and to carry through an analysis for each of these tits in turn, ignoring members wherever necessary in order that . .
the structure of each separate unit may be statically detertiate. In wing truss analysis, for example, it is the ~v-ariable practice in making routine anaIyses to entirely ignore the effect of the stagger wires and the ext ernaI drag wires, the forces acting on the truss being resolved into the pIanes of the lift bracing and the internal drag bracing and these bractig systems being designed st rongIy enough to carry the e~tire loads.
When the stagger mires are taken into con- sideration at all, it is only on the assurnpt ion that the flfig wire has been shot away and that the load must be carried from one lift. truss to the other through the stagger -wires. Obviously the members thus iggored wifl come into play under some conditiom, and, in so doing, they wilI affect the stresses in the other members which are ordinarily taken into zccount.. It is cus- tomary to fall back on the assertion that. the ordinary method of analysis is on the safe side, but reliance on such a claim is ah-a-ys unscientific and unsatisfactory, and nowhere more so than in airplane desigg, -where the Ioads acting are all dependent on the weight of the structure, and where it is therefore almost as undesirable to have one unit or group of members too strong and -.
heavy relatively to the other members as to have one member too weak, since the excessive strength and weight of one increases the loads and stresses in alf others. It is therefore emi- ‘ nently desirable that the analysis of the airplane structure should be carried through with the greatest possible refinement of detail, and that nothing should be Ieft to guesswork or chance where it can be avoided.
TO take one of the simplest cases as an illustration, it k evident that when an airplane is diving and the center of pressure is far to the rear of the rear spars the load on the rear truss wili If there were DO restraint on the relative act upward and that OD the front truss down-ward.
motion of the two systems of bracing the rear truss would therefore rise while the frod one descended below its normal Ie-wl, and the form -would be dktorted at each panel point, the truss being so warped as to decrease the angle of attack along the -wing and to decrease the stagger The physical reasoning on this point has be= given at some length by near the tips of the wing.
Mc. John Case2 Other points at which there k uncertainty are the external drag wires, already .
alluded to, and the interaction between the fuselage and wings. The latter point W% taken up in a rece~t, report of the hTatiormI Adtiory committee for Aeronautics,a but the analysis was not carried through in ftil and c=tati rough ass~ptions were made as to the temions in the external ~ag fies.
The standard method of treaktig redunda~t members and statically inde~ertiate struc- tures in generaI is furnished by the method of le=t w-ork, orimated by Castigliano. This method is commonly used in bridge de-sign, and h= feud some application in other departme”&s .
of engineering, but very fittle attempt h= FLS yet been made to apply it to theneeds of aeronautics.
The general means of %pphcation of tie method of least work will be found discussed ti any The application to airplanes has been briefly and simply discussed in textbook on structures: —— 1 The Importance of Incidence Wires b Shr@h Wmlatfom, ‘‘ hmm.utics,” Dccemb& 4, 191S.
* Fuselage stress A@i-sis, by E p. Warner ~d R. G. ~UW, Report ~0- W ~-atiOIMl Adti~ry Couttee for -%oI1311t1cs, Washington, 1923.
Mechanies of Internal Work, by Church, New York, 1910, : The Theory of Stmcturc-s, by C. M. Spotford, Chapter XW, New York, 1915.
54ss941-16 239 240 ANNUAL REPORT NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS.
Pippard and Pritchard’s recent work on airplane structures (London, 1919), and Mr. Case,4 in an extension of the article mentioned above, has treated m~t.hemat ically the theory of the eflec~ of ticidence wires by this method, but a great deal of work on the subject remains to 1-wdo~e.
The method pursued in this re~ort is some-what similar to that followed in the previous report on fuselage stresses. A representative airplane is chosen and the analysis carried through both with and without consideration of the redundancies for a number of different systems of loading, in order to give a concrete idea of the importance of the stagger wires and external drag wires and of the mtignitude of the error involved by f ailing to.take them int o consideration when FIIMlyZ- ing the stresses in an airplane of conventional type.
The stresses in each member for the various conditions of loading have then been tabulated. The airplane chosen as an illustrative example closely resembIes the JIST4H,it being probable that more Americans are familiar with the generaI characteristics of this type than with any other.
Assembly drawings of this airplane aro given in figure 1.
L- _._______T
FIG. 1.
The method of least work is really nothing more than a simple method of analyzing the geom- It-is obvious that if a structure would deflect under load, and any parti- etry of a“structure.
cular redundant tension member were absent, in such a manner as to increase the distance between the points at which the ends of that redundant member are actually attached, the re- dundant member wiH resist and reduce the deflection and will modify the strains in the other members and the distribution of load among them. (hstigliano’s theorem oflers an easy and straightforward route to the determination of this new disfiribution, whieb could otherwise be found only by a tedious process of trial and error, There are certain points which mdie it very difficult to apply the method of least work to airplane structures in the normaI manner. The end fixation. of the members is uncertain, there being an initial yield in the terminals and fittings which it is usually impossible to take into theoretical consideration. Furthermore, the stresses acting on some of the members are a combination of bending and direct end loading, and it would 4 Incidence W@s in the Strength CaIculation of Wind Structures, ‘‘ Aeronauf.its,>> December 18 and 25,1918, and January 1 and 8,1919.
ANALYSIS OP WING TRT-W sTTKEssES.
be extremely difEcuIt to take full account. of the effects of both types of stress. It appears probable that it will be safe in least -Kork analyses of the wing structure to ignore the wooden members entireIy. The tensile strength of airpIane wire is about 200,000 pounds per square inch! and its mochdus of eksticit~ is about 30,000,000 pounds per square inch. The strength of spruce in direct compression is, on the other hand, about 4,500 pounds per square inch and its moduIus of elasticity is about 1,600)000 pounck per square inch. If all the members were per- fectJy eIastic up to their ultimate strength, the strain per unit length a~ the instant before rupture would be a IittIe less &han one-half as great for spruce as for wire. 131Mhermore, the unit stress in the spruce members is always a much smaller proportion of the ultimate strength than is that.
in wires, beeawse most of the -wooden members are long coIumns of a small sectional radius of gyration, and the unit stxess mustt therefore be kept lowin order that failure maynotoeeurbybuck- ling. Since the work done in stressing a member depends largely on the strtiin imposed, being directly proportional to strain for a given stress, it is clear that the work done in stressing the -wooden members wilI be much less than that done on the wires, and that the effect on total -work and its deri-ratiws of any change in the stress in the wooden members will therefore be relatively slight. Reliance has not, homever, been placeci solely on this approximate physical reasoning.
.4D analysis has been carried through for one type of loading, taking the wooden members fully into account so far as their end Ioads are concerned, and the tabulation of results shows, as has just been predicted, that the effect of the wooden members is small enough to be negleeied under ordi- . .
Fm. 2.
nary circumstances. The comparative anaIyses with and without consideration of the spars and struts WM be ffiy discussed in their proper place.
Another question which has a considerable effect on the stress when there are redundant members is that of initial tension in the wires, and the uncertainty pre-railing as to the initial tension is of ten used as an argument against the undertaking of further refinement of the methods There is some justice in this argument for, as will be shown later, the initial of stress analysis.
tension does -vary widely between different membe~ in the same airpIane and between corre- TO show what the ma&um effect of initial tension sponding members in clifferent machines.
is likeIy to be, an analysis has been carried through -with the maximum probable initial tension* in each wire.
In the application of the method of least work to aeronautical structures there arises a problem not so often encountered in the design of indeterminate bridge structures, in thtit some It. is necessary, then, to make some assump- of the members are capable ody of tti% tension.
tion in start&~ the anaIysis as to wtich one of an Opposed Pati of te~ion members til be in tension when all the loads are acting, and then to carry the anaIysis through, disregarding entirely the members opposed to those which are behe~ed t.o carry tension in the fial result and treating the working members for the moment as though they could take either tem~ion or com- pression. If, ho-weyer, the iinal resuIt shows a compression in a wire, it is necessary to repeat the whole analysis with the opposing wire taken into com~ideration throughout in place af ihe one ANNUAL REPORT NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS.
wbieh had a stress of opposite sign to that expected. It is usuaIly possible, after a little practice, to guess which wire of my pair will carry tension, and the trial and error method just outlined therefore does not often have to be invoked.
!l%e method of least work is essentially a check method. It can not be used for iuitial calculations, as it is necessary to know the sizes of all the members before the work equations can he written. In this respect it is like the ‘(Berry method’) of wing spar analysis by the generalized ecpmtion of three moments.
The cases treated in this report are five in number, two of them relating to loadings experi- enced i]) the. air and the other three to comparison with other types of analysis and to the effects of mollifying factors. The loadings considered are those experienced at a high angle of attack ancl a high speed, -as in pulling out of a dive abruptly and in a vertical dive at limiting speed.
The other three cases deal with the effect of wooden members, the effect of initial knsion, and with the determination of the stresses encountered when the structure is Ioaded in accordance 43’7~”
r————— 1
I
I I
FIG. 3.
with a suggested set of specifications for static testing recently drawn up by the staff of the NTational Advisory Committee for Aeronautics, CASE I.
AS a first application of the analysis, the airplane was assumed to flatten out of a clive very abruptly, so that the angle of attack reached 120 in combina~ion with a speed of 100 miles per hour. The total air load uncler these conditions is 5.43 times the weight of the airplane, AcceI- cwometer tests on pursuit airplanes, conducted at the, Royal Aircraft Establishment, have never shown a dynamic load factor in excess of 4.2 under the most violent handIing, and ordinary stunting does not impose loads in excess of three times the weight. The conditions assumed at least as severe as any that would ever be encountered in flattening out of a dive.
. are therefore A perspective view of the left wing truss, ~Fith every wire numbered, is shown in figure 2.
The first step in the analysis is, as already pointed out in the introduction, to determine which wires are pIaced in tension by the Ioads being considered> as all wires which do not carr<y tension must be disregarded entirely. The possible redundancies are the stagger wires (not more than one at each paneI acting in any given case), the two external drag wires 20 and 21, and the landing It-is possible for the landing and flying wires to be stressed wires in the inner bay, 16’ and 17’.
at the same time, even though there is no initial tension anywhere, as the center section struts can carry no tension and the lift reaction on the upper wing at the center section may be carried in whole or in part b3’ the Ianding wire, being transmitted then~e to the fuselage through the inner interplane strut and tke inner flying wire. Since the point of attachment-of the Iower end of the landing wire is itself deflected upward by the normal lift load, that wire -will not take the ANALYSIS OF VKNG TRUSS STRESSES.
center section reaction if there is any other member capable of carrying ib in a reasonably direcb fashion. This is the case in the front truss of the airplane here analyzed, as the Fire 18 carries the reaction. The rear’ truss, however, is supported at the center section on+ly by the wire 19, which runs so obliquely that a relatively large vertica.I deflection of the upper rear spar at the center section would ensue if there were no other restraining member. If t,his deflection proceeds far enough, the rear landing w-ire comes into play, and it is therefore necessar~ to take this wire into account as one of the redun[iancies.
As for the two ex6ernaI drag wires, No. 20, which runs downward and forward from the rear upper spar, k obviously in tensiort, as the upward deflection of the lift. truss and the rear- ward deflection of khe drag truss both act to extend that wire.
No. 21, whiIe it is extended b-y the deflection of the drag truss, is so much shortened by the much larger movement of the lift truss that it carries no tensile stress, and is therefore clisregarded. There are, then, four redundancies in aII, incIuding the two stagger wires which are acting.
Oue stagger w-ire at.
each paneI point always comes into play, but the load maj shift from one diagonal to the other as the type of loading changes. In the particular ease under consideration it is the Iong cliagomd, running downward from front to rear, which acts at both panel points, chiefly because the front lift truss carries more Ioad than the reai, the center of pressure being far forward, and conse-
I%TE:-%4XHZ Wt!T.S MD MM8,?R5 IN F.?OYTA,?O LGWH7 /“
TRUSS .s%?w GaJ7%5.
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quentIy has a larger deilecfion. The Iong stagger ~ire accordingly comes into play to equalize the deflections. The distribution of the drag load also acts to stress the same wire, as the -wire No. 20 acts as a partial support for the upper wing at the inner panel point, and the length of the portion of the Iower -wing which is cantilevered beyond its last support (not counting the ~ stagger wires as supports) in respect of drag is therefore greater than the Iength of the corre- Part of the drag of the lower wing is therefore tramferred sponding portion of the upper wing.
.
to the upper and carried by it to the fuselage, instead of the reverse, which is generalIy assumed arid which wouId hoId true if it mere not for khe externaI drag wires.
The center section struts are incapable of sustaining any temsion, and the reactions must therefore be taken, in the nonredundant anal~-sis, by the wires 18 and 19. The horizont aI — components of the puIls in these wires combine with the center section drag truss reaction to produce an unbakmced force in the pIane of the -wing, ancl one of the cerker section struts mht k the case under discussion at present., the be thrown. into compression to take the force.
unbaknced force being to the rear, the forward strut is in compression and the rear one is inoperative. The tension in 1S is ~ery large because of the smaLI angIe -which it makes with tho for-ward strut.
The mean resultant air Ioad on the wings ~as found to be 36.6 pounds per square foot.
In &is, as in all other cases, the -rariation of unit loading between the wings and the variation ANNUAL REPORT NATIONAL AD171SOR1’ COMMITTEE FOR AERONAUTICS.
The along the spars were neglected, the load per running foot being assumed to he constant.
load was distributed between the front and rear spars in the usual manner, the center of pressure being 33 per cent of the waF back on the chord. The lift and drag reactions at the several panel points were then determined, and each lift reaction resolved into the lines of the drag struts and the interplane struts. The perfectly general method of carrying through the work would be to resolve every force into those two tines and a line parallel to the wing spars, and also to determine the direction cosines of e~ery member of the truss with respect to a nonrectangular system of axes paraIIeI, respecti~ely, to the wing spars, to the drag struts, and to the interplane struts,~ and then to write the equations of equilibrium at every point.
Having done this, the solution becomes practicality automatic. lt is possible, however, to very much shorten the work by a judicious use of the method of sections, especially if the The first part of the problem is stresses in the wooden members need not be determined.
to solve for- the stresses in all members except the redundant ones, ignoring t.hoso entirely; ‘ and this is identical with the ordinary stress analysis.
The analysis with wires 6, 7, 20, and 17’ ignored being completed, each of these, in turn, is assumed to carry a tension of 1 pound, and the stresses which every other member of the truss would bear, due to this tension, were there no other loads acting, are computed and tabulated. The total stress in any member can then be expressed by the formula:
T. =fi +-T’ Xf6 + T7Xf7 + T20Xf20 + T17’ xf17’
where T=is the total stress in the member in question, ~i the stress due to air Ioads with redun- dancies omitted from consideration, j, ~,, f,,, and ~,’, the stresses due to tensions of 1 pound in 6, 7, 20} and 17’, respectively, and T8) T7, T20, and TIT’, the stresses which actually exist in those redundant members when the structure is loaded.
The work done in elongating the member x ii ~x=T.2x] ~.~E where 1 is the length of the member, A its cross-section area, and E the modulus of elasticity Writing Tin this expression in terms of T,, T,, T20, and T,,’, of the material composing it.
every member of the structure, and doing the same for the expressions for WY, Wz, and so on, for the total work of deformation for any set of values of the stresses in the redundancies can be obtained by summation. In order that the work may be a minimum, as required by Castig- Iiano’s theorem, its partial derivatives with respect to each of the independent variables (in this case the tensions in the redundant wires) must alI be equal to zero. Differentiating the expression for total work with respect to To, T7, ancl so on for each of the redundancies in turn, four simultaneous equations in four unknowns are obtained, and those can at once be solved.
The stresses on all the members taken into account in the usual type of analysis and ordinarily considered as nonredundant can then be determined by substituting in equations of the form given for T= the vaIues just found for the stresses in the redundancies by soIution of the simultaneous equations for the work derivatives.
The carrying through of this process shows the tensions in -the redundancies to be S7 poundsin No. 6, 143 pounds in No. 7, 707 pounds in No. 20, and 1 pound in No, 17’. The important figures in connection with each member of the truss are tabulated below. Of speciaI interest are the listings of factors of safet.v as founcl by the ordinary statical analysis with alI st agge.r wires and external drag wires disregarded and as found by the complete anaIysis with these members fully taken into account.
It should be borne in mind that these are true factors of safety or “material factors,” based on the worst possible loading, and are Ims than one-fifth as large as the hypothetical “factors of safety” which are usually specified and which are based on the loading in normal rectilinear horizontal flight in smooth air.
The presence of the redundant members reduces the stress in 11 wires and increases it in onIy 3 (not including the redundancies themselves). The beneficial effect on the worst- stressed members is, however, slight.
s, — .* 6 This system of &xes would be rectangular if there were no stagger.
ANALYSIS OF _TG TRUSS STRESSES. 245 The stress in the rear inner landing wire is negligib~e and has been omitted from considera- tion in computing the factors of safety. Furthermore, the effect of 17’ is actually even a little less than would appear, as the rear portion of the fuselage is subjected to a dovmuvard dynamic load, and the point of attachment of the lower end of 19 is therefore deflected down- ward relative to the points of attachment of the lower wing spars, so that 19 carries a larger share of the upward reaction at the center section of the upper wing than it w~ould if its Iower end remained exactly fixed. The effect of the landing wires w-ill therefore be dijregarcled in alI subsequent cases. The possiijilit y of their having an effect is only mentioned as a warning that they should sometimes be taken into account, as the share of the center section load carried by the landing wires increases rapidly as the obliquity of the center section wires is increased.
CASE I.
t stress Sties StrESs nti~tl “ ~~~@ stress F. S.
due to due to due to tre#th v . 0. withctut with 81 L pound 1 pound 1 Nund ~~d~ma: 1 redund. “edund member n h’O. 6. in NO. 7. in N-o.20.
— — — lljz ... ----- . ......- ... .......... 162 2, w 16’.0 518 .6$3 . ....... .......... 577 2,603 4.50 432 ......... . .......... 5.29 .634
f% –1.038 :2 $%J 11.6
’335 .654 .054 –1.028 2M 15.1 1.033 .................... Zm 23.0 1.CQ3 ..........
........ ........ 11 14.0 – .6S3 ......... .... ....- . 198 ~% 2.57 13. I 1s0 – .683 ..................- . m 2, WI 21.6 E.% – .654 – .6s4 .......... 53$ 6, m i’. 45 – .&j4 608 – .65+ .......... 45S 4, (Ml 8.7.4 93s ...... ... .......... .......... + ~c&l 7.87 367 ........ .......... ........- . 10.9 2, b% -1.318 . . . . . . . . . . . . . . . . . . . . & g 4W 3. 5s 1,772 1.318 . . . . . . . . . . . . . . . . . . . . 1,S% 8, 4M 4.45 5,:43 -1.272 –l. m . . . . . . . . . . 4, S51 8,’03 1.73 – .3~ 3,111 1.272 1.272 3,633 8, w 2.31 2@! 1.2s2 1. 2X2 – 1.975 I, E@) 4, m 2.79 342 -------- - .......... ......... . 342 2,WI 5.85 ...... .. - .......... 1.033 707 4,m 5.94 --------- —2, 15? 1.052 ......... ......- .-. —2,062 . . . . . . . . . .......
–2, 310 . . . . . . . . . ......
–2, 174 –1. 5s3 .... .... ............
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–1,733 1.5.s3 ......... .......... –1,613 . ......
–~, 510 –; Lll .................... —2,m . . . . . . . . . ......
—a,425 .w_2 . . . . . . . . . . –5,019 . . . . . . . . . .......
-6>0s –3. @xl –1.48.5 . WW –6,355 . . . . . . . . . .......
—4 742 3.m3 1.435 — . 75W –4,739 . . . . . . . . . ..----- —4. 093 –1.979 1.179 –6,5s6 . . . . . . . . . ......
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— 214 – .4302 — . .K303 . Hz . . . . . . . - . ......
I% ......... ........-. .......... – 3m . . . . . . . . . ......
76 . . . . . . . . . . ......
31 .5301 . . . . . . . . . . ---------- 520 474 . . . . . . . . . .......
– .5301 . . . . . . . . . . ---------- – Is . . . . . . . .
– 103 1.RJI . . . . . . . . . . . . . . . . . . . . . .... ..
~$ 2, 6L’3 —2. LL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .......
2: C& . 493? . . . . . . . . . . . . . . . . . . ......
7&S –2. 036 2; S43 . . . . . . . .
3,136 – . 493? . . . . . . . . . . . ......
734 . . . . . . . .
3-B 3.033 .9s$4 . . . . . . . . . . . .. . .....
— 16~ — 162 . . . . . . . .
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– 134 .4W0 . . . . . . . . . . . . . . . . . . . . –% . .......
— 272 . . . . . . . .
–309 .4303 . . . . . . . . . . . . . . . . . . . . . .. ----- — AQ1 – 324 . . . . . . . .
.4333 .4WI . . . . . . . . . . . ...... .
— 87s –3i3 -... ..-. . .......... .......... . . . . . . . . .....-.
–62S — .7933 . . . . . . . . . . . . . . . . . . . . –m . . . . . . . . .- ......
–2, 366 .7%73 . . . . . . . . . . . . . . . . . ..- —2, %8 . . . . . . . . -------- –1,7W — .KEQ —--- (W) . . . . . . . . . . —1,887 . . . . . . . - . ..----- –$ 7S0 –1. m -1. ml 1.5S.5 – W . . . . . . . . . .......
/ CASE Ia. (Effect of wooden rnembt.m.)
The loadirw taken in this case was the same as in the last,, but full allowance -was made for the effect of~he wooden members, in so far as their end loads’were concerued, the stresses in these members and the work done in elongating or shortening being computed exactly as for the wires, and the equations of total work being enkmged to include the work which goes into storing strain energy in the spars and struts. ’11~ strain energy of flexure has not been taken into consideration, as its variation due to the redundanci~ is slight, and the accurate com- putation of flexural work would be an undertaking of great difEcuIty, requirhg a series of suc- cessive approximations to allow f or the departure of irkermediat e panel points from the straight Iine connecting the outermost and innermost supports of the.ting truss. It is only because of such departures that the work of flexure is changed by the redundant members, and these ANNUA&J REPORT NATIOITAL ADITISORY COMMITTEE FOR ‘AERONAUTICS.
redundancies therefore have no effect on the energy of flexure in airplanes which have no inter- mediate panel points, the wing truss on each side comisting of a singIe bay and an overhang, The introduction of the wooden members into the work equations gives a larger stress in two of the redundancies than was found in Case 1, whiIe the stress in the third (hTo. 7) remains pnwt.ically unchanged. The tension in No. 6 was increased to a rather surprising extent,. In only one wire (No. 18) does the allowance for the struts and spars change the computed tell- sion by w. much as 5 per cent of its ultimate strength, and the effect in that one wire, m well as in most–of the others, is to reduce the computed stress.
The effect of redundancies on the stresses in the wooden members themselves is small in most instances, but is not by my means small enough to be negligible. The loatLs in the worst- stressed portions of the wing spars are reduced hy from 15 per cent to 25 per cent l}y the redun- The stress in the intermediate compression rib in dant wires, chiefly by the effect of No. 20.
the inner bay of the upper wing, is cut clown about S5 per cent by the extcrmd dr~g wires. ‘h interplane struts are but IittIe affected, with the exception of the front cehtcr section strut} the stress in which is 64 per cent sma]ler than it wou]d be if the redundtint memljcrs were removed.
A tabulation, similar to that for Case 1, of the stresses with and without dlowanec for the redundant members is given below. The differences between the stresses found in Case 1 and Case Ia, or the errors due to failing to include the wooden members in the work equations have been included in the tabtdation.
lt appears from the comparison of the results in this case and in C&se I that the assump- tion originally made was a reasonably accurate one~mid that it is safe to omit the woodw -.
members from consideration for any except the most refined work.
CASE Ia.
[Stresses without redundancies, and eflects of writ stresses in redundancies, are the same as In (%.e 1.]
- II Differ. DRer- D1fier- I ence ence erwe Stress Stress between Stress between }etween No.
No. with all [s and I NO. wlt.h all Ia and I a and I redund. absolute redund. (ab~lu te %%%! I absolute magni. mJ..i- magni- tude), tude).
,.
d----
162 1,186 –31: 35 . . . . . . . ....1 113 + 37 1. . . . . . . ..
2. . . . . . . . . 62.f + 4! 342 36 . . . . . . . .... 437 -1-37 3. . . . . . . . . 539 + 47 911 + 204 37 . . . . . .. .... 1+38 4. . . . . . ...1 1S3 –163 –1, 9s9 — 73 38.. . . .. . ... 2, 2:? -146 5. . . . . . ...! 102 –162 –2, 420 ~ + 110 39 . . . .. . . .... 1,262 + 191 —.
6. . . . . . . ..J 155 + 68 ‘1,~1 i -106 40 . . . . . . . .... 2.,662 -181 7. . . . . . . ..t +215 – 47 –’i, 803 –216 42 . . . . . .. .... – 162 – 46 –6, 473 -I- 108 43.’. . .. .. .... – 67 – 2: - 43 –4, 667 — 72 44 . . . . .. . .... – 242 – 30 lki:z 1 :: 414 - 44 –6,550 — 30 45 . . . . . . . .... — 296 – 2: 11. . . . . . . . .
50s 0 – 299 ,$:: 46 . .. . . .. ....
12. . . . . . . . . ~ y’; 367 0 – 36!2 47.. . . . . ....
13. . . . . . . . . -1- 55 2,251 + 30 4a. . . . . ..... –2:243 – 55 14. . . . . . . . . ! – 587 1,977 ;%’ – 88 .!9 . . . . . .. .... –1,942 + 55 15. . . . . . . . . + 126 4>767 –64 – 320 0 50. . . . . . . . . . . – 6<8 –303 16. . . . . . . . .
17. . . . . . . . . 3,621 – 12 I _!. _ I_.
-.
1 The stress is changed in sign in this case.
CASE 11, The loading in this case was that encountered in a -rerticaI dive at 120 miles m-w hour- This is consider~bly below the limiting speed of the JN, but is as fast-as it is likely to ~e dived.
It was assumed that the upIoad on the rear truss was equal to the clown load on the front truss, and the resultant force on the wings was therefore pwdIel to the chords. Since the an”gle of attack was negative, there was some lift on the wings under these condii ions, but not enough to balance the down load on the tail. Jn the particular machine used as an illustration the angle the zero lift angle for the Eiffel 36 section being unusually small, of zero normal force is —5°, The true angle of attack in a vertical dive would probably be nearer – 4° than –5°. I’he com- ponents of load acting perpendicular to the wing chord were 45 pounds per foot, gi-ring a total fcrce of about 1.6 times the weight of the airplane in each lift truss (including both the right ‘ ANALYSIS OF WING TRUSS STRESSES. 247 and left s;des of esmh truss).
This is unusually large, the Eiffel 36 wing having a~ exceptionally large diving momenti at the angle, of zero lift.. The loading in diving the JN to 120 miles per ho= is abo-ut equal to that wti~h -would be found at the t&ninal v~ocity with most airpla~es The load parallel to the wing chord (front and UShg the R. A. F. 15 or other similar section.
rear trusses together) was 7.22. pounds per foot, so that the total distributed load on the drag trusses, including the parasite resistance of the interplane bracing, but not including t-he components in the planes of the wings, due to stagger, of the lift truss reactions, was about 29 ln a dive to the terminal _relocity this force may rise per cent of the weight of the airplane. —— to as much as 50 per cent of the weight of the airplane for a machine with tie lines and low parasite resistance.
The front king-post bracing abo-re the upper wing is stressed by the down-load on the front, truss, and the stresses in the two lift trusses therefore are not quite s-ymmehical with respect to each other. ~f the two systems of trussing were parallel throughout, the stagger would have no effect on the net. reactions in the plane of the wing, as the effects of the inclina- tion of the Iift bracing would be equal and opposite at the front and rear paneI points and -would exactly canceI out; but this is not actually the case, since the king-post overhang bracing lies in a plane perpendicular to the wing chord instead of being paralleI to the lift truss proper.
There are three redundancies in this case, Nos. 6’, 7’, and 21. The stagger wires acting are those which run upward and to the rear, as mQht be expected, since the rear truss tends to move up and the for-ward one down, and the stagger wires acting are those which are thrown into tension in resisting this relative displacement of the lift trusses. The work equations were -!?4 pounds in 6’, 427 pounds in 7’, find 485 pounds in 21. lt might perhaps have been antici- pated that No. 20 would be in tension, as the rear truss considered alone tends to move up-ward and to the rear and both of these components of motion would eIongate No. 20, but. analysis shows tha.b hTo. 20 would carry a considerable compressive load if it were capab~e of sustaining — such a load. The physical explanation of this is dual. In the first place, the pull in stagger wires hTos. 6’ and 7’ tend to draw the upper wing forward. SecondIy, and more important, the load in the rear truss is carried by the flying -wires, -ivMe that in the front truss falls on the landing wires. These, being single in each bay, elongate more under a gi-wn load than do the double flying wires, and, if the two trusses -were not connected together in any way, the front one -would deflect dowmward. more than the rear one vould yield upward. Since the two are connected by the stagger wires and must mo-re substantially together, the effect of the dis- symmetry of the lift and antilift bracing k to cause the wing cell to deflect downward as a whole.
The upper rear spar therefore mo-res, not upward and backward as it would if there were no redundancies, but forward and downtvard. Incidentally, this ser-res as an excellent illustration of the intricacies of a redundant structure and of the manner in -which the stress in any member depenck on the form and strength of ever-y other member. For example, if the antilift wires as -weIIas the lift wires, were double there is but lit t]e doubt that the upper drag wire (h~o. 20), as well as the lower one, would carry a considerable tensile load during a di~e instead of going sIack.
The pull of the stagger -wires, drawing the upper wing forward, also has the effect-, not very generally foreseen or allowed for, of throwing a Ioad on the antidrag wires in the upper wing.
A load on these wires is ~~pected at large angles of attack, particularly in airplaries with little or no stagger, bu~ its appearance in a vertical dive seems rather curious until a thorough analysis is made.
The nature of the load distribution in the center section is quite different from that. at. a large angle of attack, aIthough three of the four members involved are active in each case. In a dive, the front wire (ATO-1S) takes no load. Both struts are in compression, and the forward tendency of the upper wing, due to the pull of the stagger wires, is resisted by a tension in No. 19.
A tabulation of stresses, similar to that g-hen for Case T, appears below. There has been no recomputation of redundancies with the work done in the spars and struts taken into account in this case, but the final stresses in the wooden members ha-re been computed with allowance for the redundancies found by writing the work equations for the wires alone.
ANNUAL REPORT NATIOhTAL ADVISORY C:O.IIMITTEE FOR .4ERONAUTICS.
The fzctors of safety in the wires are high and fairly uniform. The stresses in the wooden members, with a few exceptions (chiefly the internal drag struts), are reduced by the intro- duction of the redundant wires. This is particularly true of the worst-stressed portions of the spars, the maximum direct compressive loads being reduced by about 72 per cent. It is a curious fact that every bay of every spar is in compression in- a dive, the effect of the stagger wires and of the king-post bracing being sufficient to overcome the tension which might nor- mally be expected to appear in the upper front and lower rear spars.
The stagger wires are of enormous benefit as regards the lift trusses. In the lift and anti- lift wires, as in the spars, the stresses are from 55 per cent to 70 per cent lower than theY ~~uld be if the stawzer wires were removed.
CASE II.
r Stress stress F.S. F.S. ! Stress Stress F. s. F. s.
No. without ~ith alI tithout _sith No. without with all without with redund. edund. ‘edund. redund. reduud. mdrrnd. rednud. redund.
— — l—l—
—l—1
1 . . . . . . +133 19.6 19.6 25....... –1,222 + 133 2 . . . . . . . . . . . . . 36.3 . . . ..i. i< 26.. . . . . . +1,225 + 72 2’ . . . . . . . . . . . . . . +289 . . . . . . . . . 27 . . . . . . . –3, 094 . . . . . . . 22.6 . . . ..i.i6. 28 . ...=..
3 . . . . . . + 11.5 +1,375 +245 29 . . . . . . .
3’ . . . . . . . . . . . . . . . . . . . . . –3, 255 . . . . . . . 20.0 . . . . . . . . . . 30 . . . . . . .
4 . . . . . . +–200 – 45 ‘t’ . . . . . . . . . . . . . . +488 . . . . . . . . . 4.10 31.., . . . . – 73 5 . . . . . . + 240 . . . . . . . 16.7 . . . . . . . . . . 32..: . . . . – 131 .5’ . . . . . . . . . +447 . . . . . . . . . 4.47 33 . . . . . . . – 158 . . . .
6’ . . . . . . . . . . . . . . +424 . . . . . . . . . 4.72 34 . . . . . . . I – 976 7’ . . . . . . . . . . . . . . +427 . . . . . . . . . 4.68 35 . . . . . . . -!- 90 8 . . . . . . +415 47.6 6.27 36 . . . . . . . – 933 + 55 9 . . . . . . +455 2$.6 5.72 37 . . . . . . . + 17 -1- 94 lo . . . . . . +457 28.2 8.75 38 . . . . . . . –2, 581 + 142 H...... + 182 +497 22.0 8.05 39 . . . . . . . + 751 12’ . . . . . +513 3.90 3.90 40..:.; . . –2, 474 i- 513 12”. . . . . 3.86 3.86 41- . . . . . . +– 614 .+ 518 +518 13 . . . . . . +211 19.9 19.9 42 . . . . . . . – 34 +—211 14’ . . . . . + 528 3.44 7.96 43 . . . . . . . – 59 +1,223 -i-359 7.98 23.1 44 . . . . . . . – 93 15. . . . . . + I, 053 +770 1.90 I 5.46 45 . . . . . . . – 119 16’ . . . . . -!-2, 206 +860 3.81 46 . . . . . . . – 530 17 . . . . . . +2,206 9.77 7.9’4 . . . . .i:ig 47 . . . . . . – 377 18 . . . . . . -1- 529 . . . . . . . .
48 . . . . . . – 999 19 . . . . . . + 202 -!-5U2 9.9) +485 49..:.+. –1,015 21 . . . . . . . . . . . . . . . . . . . . . . . . 8.67 22 . . . . . . – 410 -632 . . . . . . . . . . . . . . . . . . . 50. . . . . . . – 625 33 . . . . . . -1,133 –523 . . . . . . . . . . . . . . . . . . . 50.k... . . . . . . . . . . .
24 . . . . . . – 354 –82Q . . . . . . . . . . . . . . . . . . -
t I !
CASE 111.
The loading in this case was one devised b~’ the authors awl recommended for USGaS a .- .
standard in sand-load tests. It was based on an attempt to distribute the load o~er the wings in such a manner that both lift trusses and both drag trusses would simultaneously roach the The total load on the wings was taken as worst load which they ever encounter in flight.
5.3W. The center of gravity of the load was placed at 37 per cent of the CIIOId from the leading edge, and the chord was assumed to be inclined at 6.50 to the horizontal, the trailing edge being lower than the leading edge (the wing truss, of course, being inverted for sand-load test}. The load per running foot was 84 pounds in the fro~t truss and 78 pounds in the rear.
The soIution was exactIy similar to those for Cases I and II and calls for no special comment.
Since the load -was nearly equally distributed between the front and rear trusses the stresses in the stagger wires were extremely small, cliffwent diagonals being stressed at the two panels and the stress in the shorfi diagonal at the outer panel point being less than 1 pound. The larger pull in the long stagger wire at the inner pa~le]~~oint iS due tO the forward reaction ‘f the Both externaI drag wires carry somo upper external drag wire on the upper wing at that point.
load, the upper one takiqg more than the 10WW, asjthe upper wing, ckfkcts more freely in the direction of the drag truss than does the lower and M the upper drag wire also assists in carrying the M.
The nature of the stress distribution in the redundancies causes a very peculiar reversal The direction of the load- of direction of stress in the internal drag br~cing of the upper wing, carrying diagonal reverses twice, so that the Ioad-ca:rying members. are arranged as in a Warren The compression ribs at the poinbs where these truss, but with all the members in tension.
ANALYSIS OF WIIITG TRUSS STRESSES. 249 re~ersals occur carry no load at all, and a sand load in accordance with these specifications would therefore bo unduIy easy on the upper drag truss in the inner bay. The stress in the upper front and lower rear spars, also, are considerably less in Case III than in Case 1, par- ticularly in the inner hays. The drag wi.w in the inner bay of the lower wing znd some of the compression ribs in both upper and lower wings, on the other hand, are stressed more severely in the sand load than they ever would be iri flight. The comparison of the results of the various analyses serves to emphasize the impossibility of devihg any si@e sartd load which -will tmdy simulate all of the C( -worst, conditions” that may be encountered in the air.
CASE IU.
I titl-ess stress F.S. stress stress F. S.
No. without tith all withW No. without mm au with all redund. redund. redund. redund. mdund.
%dmd.
3ss. 392 6.63 26.. ---------------- –2,1s2 –1,$23 L.........
;-:.::::::::::.
S35 3.11 27----------------- –%,353 —5,870 1-. ........
346 2. ra 2s . . . . . . . . . . . ..--.. - —2,1&2 –2,154 f. . . . . ..-.
2. ~~ ; !+:::::::::::! _l,Fi —7,169 L . . . . . . . . .
29. . . . . . . . .- . . . ...-.1 -s>6S4 442.2 ~ 3ss 1-.. -.... - 5. . . .._. -.-.-\ I,ms 2 30. . . . . . . . . . . . . . . . . . – 3SS . . . . . . . . . .
6’. _. . . . . . ------ — 1 3$3J 31. . . . . . ..-- . . . . . . . . – 561 – 561 . . . . . . . ..- ‘i .....-.-.-..._. ......... 32. . . ..- . . . ------- + 1.% — SS7 $ . . . . . . . . . . . . . . . . . 396 ?2 6: S$ 23. . . . . . . ---------- — 35 ; 23ZJ . . . . . . . . . .
9... -_... ----- 497 5.22 34......... -.–_-. + 235 . . . . . . . . . .
10.. -.-. . . . . . . . . . . l,4&l SJ31 5.02 ~ 35.. ---------------- – 70 . . . . . . . . . .
11. . . . ----------- l,w2 4.43 36. . . . . . . . . . . . . . . . . + 562 ;2 ---------- ‘ 12. . . . . . . . . . . . . . . . 404 % 37. . . . . . . . . . ..-.-.. —436 – 457 . . . . . . ..- 13. . . . . . . . . . ..--.. 376 376 38. . . . . . . . . . . . ------ +2,50i’ +2:33% . . . . . . . . . .
14. . . . . . . . . . . . . . . . 1:954 1,S5.5 39-. ..- . . . -------- + 175 + 390 . . . . . . . . . .
lo-.. -.-. ..-..__ 1,819 1,818 4.62 40. . ..- . . . . ..-. ----- +3,327 ~ +2,941 1 . . . . . . ..-.
. .
16---------------- 4: C93 3,879 ~&...--.---_—-.. —i24 –2$0 . . . . . . . .. .
– l% . . . . . . . . . .
lo . . . ..- . . ______ 3,au 3,217 4,362 1,35S :::::::::::::::; 1;:; – ~s ,:.:------- 1 1o---------------- .
— 19.. - . . ..-. -.....-! 351 351 5.70 -. . . . . . . . .
.—— — Zo. . . . ..- . . . . . . . ..l---------- 1>620 2.5’2 45.. . . . . . . . . . . . . . . — 7X 555~------- — 21.. . . . . . . . . . . ...-[---------- 207 . . . . . . . . . . . . . . . . . . . . 6991 .........
–1,50s H. . . . . . . . . ..--... -E! — 650L .........l 22. . . . . . . . ..’...... –1,5GT 33 . . ..-- . . . . . . ---- –2>644 —2, 643 4!.. -.--.. - . . . . . ----- _1,7m +; /.::. ......
‘:::::’:: 1’ 24. . . . . . . . . . . . . . . . – 357 –35.S 49. . . . . . ..-_. ------~ . ....... .
-------- .
!
2.5.. . . . . . . . . . . . . . . –3,3s!3 –3,379 50... ---... ---..:–3;251 – ‘3s7..........
--------- j EFFECT OF llQXHAL TEXS1ONS.
The anal~=es of the&t two cases have been based on the assumption that all of the wires are just taut ‘but with no initial tension. .lct.wily, e-ren if it were possible to secure such an adjustment it would not be desirable to do so, as some initial tension is necessary in order to keep tihe structure from vibrating badly and to hold it in proper alignment. It is therefore necessary to inv-estigate the effect of initial stress on the distribution of load.
It is not correct to apply the method of least work in a straightforward manner, taking the derivatims of the -work done by the external loads along, or of the change in total strain energy due to the imposition of the external loads, as might at first be assumed to be the case.
The partial derivative of the tottal strain energy with respect to the stress in any member is equal to the deflection, parallel to the line of that member, of the point at which the force representiQ the stress is considered to be applied, this deflection being measured from the point at which there would be no stress in the member in question. If the frame of the struc- ture is Iined up with initial tensions in some or aIl of the members, the deflections which are de~~ed in order to ~tabhh the conditions of geometrical equilibrium of the truss are those measured from the strained Iengths of the members before the external loads are appIied, and it is therefore necessary to make a deduction for the initial deflections due to straining of the redundant members against each other. The equations based on the- -work derivatives, and defining the relations between the final stre=es in the redundant members, must then be written: dP7 ~W ——- .
d T= Clt= 0 where Tk the work done by extermd loads, w the work of deformation when the initial stresses alone are acting, T= the fid tetion in any redundant member and & the initial tension. It is not necessary, howewer, to re-write all the equations, as it k sufficient to carry through the 250 mm7im REPOFm NATIONAL ADVISORY COMMITCEE FOR AERON.4TJTICSL analysis and compute the stresses without regard to the initial tensions, and then to add to the stress in each member that due to initial stress in the redundmcies. It is evident that this is the case, as the equations for W and w in each member are homologousj e~cept that the terms involving only one unknown stress do not appear in the latter, since those terms are due to the external loads. The derivatives }~and $ are then identicd, except that the second invol~ed x z t where the first has T, the subscripts remaining the same, and that the first– has a pure numeric al term which is lacking in the other. The terms combine, when tlie second expres- sion is subtracted from the first, in such a way that neither T nor t appears singly, but always in the combination ( T–t), and it–would therefore have been sufficient to write in the fist place dTr .
d ( Tx–&)= U ( T-t) 21 (T’-t’) 1 where 117 is given the fictitious value 2A ~ instead of ~~— j which is the true cl ~ wlge in The solution of tile sirnultcmcous strain energy caused by the application of the external loads.
equations then gives T.–k for the redundant members, and the initial stresses must be added in to secure the total final load.
The effect of initial tension can best be illustrated by giving CL couple of simple examples.
As a first-instance the pin-jointed structure shown in figure 3, md consisting of bars cross It is_assumed that the bars are so large in proportion to braced with wires, may be selected.
the wires that their strain may be neglected, and that the two diagonal wires are of ecpm] size.
If an iriitial tension F be placed in one wire there must be an equal and opposite initial tension resisting it in the other dia,gonal member in order that the structure : may be in ecluilibrium. If an external load 0.707 P be applicd as p shown in the figure wire No. 1 will carry a tension of P pouncls while \ + No. 2 goes slack if there is no initial tension. If there is initial tension No. 2 will shorten by exactly the same amount, that-hTo. I lengthens, P and the resultant tension in hTo. 1 will be F+ ~ ~ while that, in No. 2 m is l?– ~. The tensions will varv in this manner as P-is increased until FIG. 5, P – 2F~ at which time the tensions are P– and 0, Thereafter the stresses are the same as if there had been no initial tension, If this very simple problem had been treated by least work with initial tension the stresses determined would im~e been+ ~ for 1 P and –2 for 2. Adding these stresses algebraically to the initial tensions in tfle two membws the same result is obtained as was just gi~en as a result of elementary geomc.trical reasoning.
If, in this problem, No. 2 had only half fihe cross-section area of No. 1 the initial tensiom .
in the two would, as before, be equal. An applied load superimposed on the original stresses wouldj however, procluce twice as great an effect b 1 as in ~, since the increase in teusile strain of 1 as the structure deforms must be equal to the decrease of strain in 2. The unit st.rcsscs in the two are then equal if they are of the same material, and the tohd stresses are proportional It follows from this that the total loads in the two wires are ~iwm, to the cross-sectional areas.
... , 2P P so long as they both remain in tension} by the formulae F+ ~- and F—~ ad that the lightcr wire will not become slack untiI P = 3F’.
TO afforcl some indication of the initial tensions e.xk?ting in airplanes rigged in tile field under average conditions and without using a t.ensiometer, tensiometer measurements of the stresses in a.11 the exposed wires were made for 6 JN4H airplanes, four of them rigged by four The averages are tabulated clifferent Army crews and the remaining two by a civilian crew.
below, together wi~h the “mean deviations showing how widely the tensions in corresponding In the case of the flying wires, the mean deviations given wires -varied in the several machines.
ANALYSIS OF WING TRUSS STRESSES. 251 are the mean deviations of the total stress in the two parallel wires from the mean value of that total, and the figures in parentheses, immediately under those mean deviations, are the means of the differences between the tensiom in two parallel wires on the same airplane. The tensim- eter readings taken in this way do not directly represent the true initial tensions, as the weight of the ceLlule is an external load which was being carried by the landing wires at the time when these measurements were made. The tensions read in the landing wires were therefore a little higher than the true initial tensions, while the values for the flying -mires were correspondingly too low. This effect, amounting to about 60 pounds in some wires, has been corrected for in compiling the table of means. The magnitudes of the mean deviations in initial tensions strongly indicate the advisability of using a tensiometer and straining all wires in accordance with a schedule specitled by the builder of the airplane. This method has been tried in rigging one or two machines at La@ey I?ield, the t.ensiometer being used by mechanics with no pre- vious experience with such an instrument, and a great improvement in the rigging was mani- fested. Where it had been common for one or more wires to tibrate badly at all engine speeds when the initial tension was adjusted by feel in the usual manner, there was no vibration except at one critical speed on the machine rigged by t-ensiometer.
It has been assumed thai the probable maximum of initial tension in any particular wire given reasonably competent and careful rigging, is equa.I to the mean of the tensions for the six machines examined plus ttice the mean deviition. This is not by any means an absolute maximum, and it was exceeded in some wires on se-rera.I of the airplanes examined, but it represents a figure which Need not and should not ever be exceeded. These probable maxima have also been inclucled in the tabulatiori above. In the case of the stagger -wires, where both wires remai~ in tension and ii is ordy the amount of unbalanced tension or the dfierence between the two, which must be t ~~keninto account, the assumption in the analysis has been that ihe wire stressed by external loads (the long one) has an initial tension equal to the aver- age for the six airplanes plUS & mean deviation and that the short wire carries a stress less than the average b-r an amount equal to the mean deviation for that member. The difference between t~e t~~o is therefore t&ce the average of their mean deviation.
TABLE OF MEAN INITIAL TENS1ON ON’ SIX JN4ES.
kobabIe Awmge Xean Probable Wim~O I $x~~d~~~n.
Tie h70. . . maxi- tension. de~ iation L%?irnm mum.
1.22 923 ,5... - . . . . . . . . . ..[ 031 UN W5 678 ~5,.:D::by~::::: !. . . ..i%.. (23] ---------
1% 1,077 1, 16C
105 707 16. . . . . . . . . . . . . ..- 64.5 11s 381 %4 (Double) . . . . . . . . ..6jj.. (~? . . . . . . . . .
(38 16’ . . . . . . . . . . . . ...1 1, m . . . . . . . . .
333 17. . . . . . . . . . . . . . . . 192 1, 06s 161 17,- f~~l.e~::::: ._..;;.. (54J . . . . . . . . .
(2$j . . . . . . ..- 305 18--------------- 466 105 676 138 19. . . . . . . . . . . . . . . . 630 n W* $97 (M~ 20. . . . . . . . . . . . . . ...” 225 31 2S7 . . . . . . . . .
% 441 1,114 Al. -- . . . . . . ..._\ 269
1.7
, It I [ The differences between the initial tensions in any given pair of opposed wires can be com- puted, if the initial tensions in the redundancies are known, on the usual assumption of fric- tionless pin joints. Any discrepancy between the difference of stress thus computed and that found by actual measurement is then due to the partial rigidity of the joints and the continuity of the spars. If, -when the structure is in perfect alignment, there is a difference between the computed and measured stresses in the nonredundant members, it shows that the wings are .
warped and that they hz-ve had to be initially stressed to draw them into alignment. In the average of the six machines measured this discrepancy was largest in the inner bay of the rear truss, where it amounted to a deficiency of about 200 pounds in the temion in the flying wires.
This is largely due to the relative bo~~ of the left rear spars in order to give “droop” to that wing and balance the engine torque.
25!2 ANNUAL REPORT NATIONAL AD171SOR~ COl&MITTEE I?OPL.4ERONAUTICS.
The effect of the maximum probable initial tension has been computc~ for all tl.mee of the loadings thus far treated, and the results are tabulated below. Tn general, the effect on the -worst-stressed members is injurious, and the initial tensions should therefore be kept as small m possible -withoui permitting excessive vibration. In tabulating the stresses due to initial tension it has been assumed in every case that the e_~ce_ss tension is in tha~ stagger wire where it will increase the stress, as both stigger wires of an opposed pair m-e in tension at all times with .
It will be noted that the factors of safety in the stagger wires me lo-w, the usual initial tension.
as their initial tensions are a large proportion of their ultimate strengths. The change of tension in the stagger wires under load is therefore small in comparison with the initial tension, the stress in one wire increasing while that in the other decreases so that. the change in each wire is equal to approximately half the tension computed by the least work analysis. q’his is in accordance with the results of sand load tests, where tensiom.eter rnemurernents after the application of each load have shown that the s~resses.in the stagger wires vary only ~ little from their initial values. In addition to al-ways taking the worst condition as regards the initial distribution of load between the stagger wires, the stresses in the external drag wires have been taken as the probable minimum, instead of the probable maximum, wherever that would be the worst condition as regmds the resultant stress in any particular member.
In a few cases the influence of the initial stress is great enough to control the direction of the diagonal which carries load in the internal drag bracing, the load shifting from the drag to the antidrag wires, or vice versa, if the excess unbalanced tension is transferred from one In some cases this leads to difficulty where the worst stagger wire to the opposed member.
loads in the spars and in the internal drag wires occur under different conditions of initial adjustment znd where the worst load in the spars corresponds with a reversal of stress tmd a transfer to the opposite diagonal from that which normally carries the tension in the internal truss. When this occurs it would be necessary, in order to secure stlictly accurate results, to carry the whole analysis through from the start with the antidrag wires included and the dmg wires omitted, but a close approximation can be made without-the necessity of repeating the work in this manner. This approximation is based on the assumption that a compression in one diagonal of a rectangular frame can be replaced by a tension in the opposite diagonal, an assumption which would be true if the frame -were exactly symmetrical and if the drag and antidrag wires were of the same size. 1f any particular combination of initial tensions gi-res a negative resultfor the total force in a drag wire this wire is therefore replaced by the opposed member, and it is assumed that the result ant stress determined is unchanged in magnitude but reversed in sign.
A correction has to be applied to the stresses in the spars in the panel where this reversal occurs, as the drag and antidrag wires do not affect the same portions of the spars. l“n the second panel from the tip of the upper wing, for example, the stress in 22 (see figure 1) is M ected by a force in 2’ but not by one in 2} whereas exactly the opposite is the case -with 23. It would therefore be necessary, in arbitrarily passing from 2 to 2’ as the load-carr34ng member, to sub- tract (algebraically) from the direct load on each spar panel an amount equal to the component parallel to the transverse axis of the stress in the wire.
The correction is subtractive in each case, as there is taken away from 23 a tension due to the fictitious compression in 2, while there is added to 22 a compression arising from the real tension in 2’, this tension being equal in magfitude, as a]ready n.ottd, tO the theoretical compression found in 2, ?-n the tabulation, wherever an approximation of this sort has been made the stress for the member aff ectcd is placed in parentheses.
In the members (interplane struts and compression ribs) directly interposed between two points of attachment of stagger wires, the fact that both wires remain in tension under dl conditions has been allowed for. The final stress in. any stagger wire is approximately equal to the initial stress plus or minus half the computed stagger wire tension (the stress being increased in the diagonal which was originally assumed to be strcssedj decreased i~ the other). This, again) is only an approximation, but approximations are ess~tial if the work is not to be complicated beyond all endurance by the introduction and simultaneous treatment of about 20 redtmdancies.
!253 INITIAL TENSIONS, CASE L Stress Stress stress stress stress Stress Stress Total Totsf due to due to :-1: due to due to W&& . . .
without No.
remltant Xo. . initlrd I.mtial initial initizl restitant init id stress. kll.?mn temon tension kerlsson tw.=krl stress.
tensson. knrion.
in 6. in 7. in& in i.
[ — ~ I .
l.......; 163 27.......;. –5,307 0 –1,040 –627 312 –6,64?2 o o 774 4,739 2. . . . ...1 :: 0 – 691 —?64 –2,1’s1 :y# 197 0 2s . . . . . . . . .
3. . . . . . . 432 0 –1,1s3 436 679 2!? . . . . . . . . . –5,52s –s33 197 d . . . . . . . . ~ ‘jw& 644 30. . . . . . . . . – 269 343 276 – 395 o 0 189 –l&l .5. . . . . . .
563 31. . . . . . . . . – 332 264 276 — lx 0 189 –164 1—1, 11~ 6. . . . ...1 1 %56 32. . . . . . . . . — 5s7 0 — W –46: 923 o .
1. . . . . . . — ~~ _182 11>147 33.....:.:. – 214 1: I, 077 14 — m 0 – 320 – 3%) 8.. --.1 19s 0 o 0 13: 0 329 34. . . . . . . . .
i6 9. . . . . . . 0 0 131 0 251 35. . . . . . . . . 15!
10 . ...... % 778 34- . . . . . . . 474 % 11: 102 0 0 126 11. . . . . . . .
700 116 126 ( ‘[~) o 0 37.. .-... - ......... — 1s 50s % 0 0 13 . ......1 2,437 367 38. . . . . . . . . I 367 0 406 2,843 .0
2;594 39. . . . . . . . . 2,341 0 753 2,032 24 1, m 2A 0
2,843 2,267 40. . . . . . . . . 15 . ...... 1,ss6 ml 3,431 381 0 0 ~k 734 5,321 41. . . . . . ...! 16 . ...... 4,s51 2 ,:2,c-&) 4E 0 4,453 42. . . . . . . .. I 17 . ...... 3>633 537 –H 36s – 162 o –: 0 2, o% 43. . . . . . .. . . . 18 . ...... 1,505 541 –s3 — 17’2 371 – 96 –322 0 19 . ...... 342 0 –s3 — 701 0 – 272 0 :&It 44. . . . . . ...1 –37: — 324 20 . ...... . 677 0 –s3 – 4s3 22.: 0 964 45.. . . . . . . . — 76 22 . ......1–2,M2 –& —3 264 46.. . . . . . . . 0 –6il 11,514 – S7S 0 0 0 k? 23 . . .. ...!–2,310 -45s —Z: 76S . . . . . . . . . . ; 0 –734 11,401 — m2 0 0 1—3, 045 24 . ...... . –1,613!
–1, 917 4s.. .-... -. 0 –153 –w —2, 29S –6A 0 0 25 I—2,917 —3, 303 49. . . . . . . . . –230 –611 –1, SS7 –s%! 0 m:::::::! ~j:~ 25!
–12415 (–5, 624 ) al._ . . . ..i —17; –m5 –597 – ’348 430 0 1’ INITIAL TENSIONS, CXSE IL n I stress Strrss stress Str es stress stress due to due to Total due to due to Totaf I!Fe?o ~ without mmmrrt indird No. initiaI initiaI i initisl No.
initial resultsnt r~$lllllt initfal init id tensson tensfon tensson L5& tension strers.
tension.
tension.
in 7’. inr. I in 6’. in 21.
I
. — 1. . . . . .
– 599 –1, 383 o 133 28. . . . . . . . . —174 o 0 –610 % 0 –416 2’ . . . . . 13: 0 –I, 03$ ~l(–z.m) m 29. . . . . . . . . –629 0 0 30. . . . . . . . . –225 3’ . . . . .
–S@ 377 o 0 I:;ti 2? o 4’: . . . . 126 u: –S3 729 31. . . . . . . . . –1s0 0 0 ~,.e--e 447 o 326 116 –33 6s9 32-. . . . . . . . –379 -376 0 /: g 424 o 6’ . . . . . 492. 0 –83 Ino 33 . . . . . . . . . –32Q — 76 0 7,cz-.~ 427 o 497 0 –so-4 Znl 34. . . . . . . . . –421 0 0 – 102 415 o 8 . . . . . . 132 0 —m 547 35. . . . . . . . . –1SS 0 0 455 9. . . . . . 132 0 – 45.2 – 557 .5&3 36. . . . . . . . –101 0 0 10 . .. ...
457 – 7: 126 116 –205 — 743 624 37. . . . . . . . . –539 0 ’44: – 74 11 .. . . .. 1.26 f.m –394 (:J$:) 664 38 . . . . . . . . . –s25 –32: 12’ . . . . .
o 0 o
–50.4 r 513 39 . . . . . . . . . –s97 –%
13. . . . . .
–1, (B$ (l:,C# 51s o 518 40. . . . . . . . . -WI 0 –m –2 14’ . . . . .
528 o 34 0 – 529 m 41-------- –972 -w —0 J12 — 1.5 . . . . . .
– 3EJ3 359 739 42 . . . . . . . . 3s0 <w –259 o 16’ . . . . .
770 – 1: 1.6.57 43. . . . . . . . . –2s4 367 ~: –s3 – 367 8 SS4 1,766 44 . . . . . . . . . 367 –s3 —1646 –303 —376 4: — m –: K.”.:: 109 m .83 – 436 7CLS 45. . . . . . . . . –3243 76 49 1- 1>~ 441 21 . . . . . . o – 734 926 46 . . . . . . . . –530 0 0 –&? 22. . . . . . -14 0 – 734 1– 932 – 734 47. . . . . . . . . –377 0 1— 1,455 –523 8 — s26 4s- . . . . . . . . 23 . . . . . . –204 — 229 –586 –8.4 0 –~~ – ~2g 1— 1,4~ –.sm 24. . . . . .
o –1, p 49. . . . . . . . . —5% –356 : 0 –30U 0 2s . . . . .
– 2s7 – 1, 1S2 ‘– (56 50. . . . . . . . . –615 –34 —456 .
—253 M......
–65 22 0 –644 E&4 . . . . . . . –210 –lx! –504 –3$ –z -ss —1 27. . . . . .
o –2, Oils –s94 421 I The. maximum probable stress here is nob equal to the sum of the tigures in the first four ml-$ as the sttess i ., :,..
.hthtigO~k rtihtetion, v~i~o~yh~=--.~’’’~ --- . ... . . . . . . . . .. ,., ,, ..,.-, ~......,~..., -,’ m one stagger wire of exh psir. The stress in fnter ... ., ., .,. , ., inteqosed, is therefore less, in mmt -, WIWR t ,, .,, ,-. -, J. ._. -... ... -. ----- .u .,_. .. . ....=. ._ —. —-.:. .;. , ,“ ,., .. . ..
!other compeneni of stress from sn entirely difkent S&me- fmum probabIe initial knsiorL foimd by measwremcnt. CuIO- as any initiaI tension in 19 wrx.rld hve to be balanced by a msIy2is.
~er wire carries an excess of initim tension.
.4NNlJAL REPORT 3TATION.AL ADVISORY (IOMMI!TTEE FOR AERONAUTICS, CASE 111.
—,; I Stre$ stress Stw;s Stress stress stress ‘iritho ut without without No. worst No. worst worst No.
initial initial initial initial initial hi tial tension. tension. tension .ension. ,emion. tension.
—11 k————
392 392 lS .......... 1,358 1, S86 35. . . . . . . . . . – 70 – 172 1,033 19 .......... 351 38. . . . . . . . . .
.335 351 562 – C&4 1,J44 20.......... 1,620 1,937 37 . . . . . . . . . . – 457 – 660 ;: :“ 207 38. . . . . . . . . . 2, 33s 2,664 –1> 508 –1, H! 39. . . . . . . . . . 1,009 9!
, 67s –2, 643 –3, 103 40. . . . . . . . . . 2, M 3,393 -1,103 2Qi 1, ?s7 %:......... - S58 41. . . . . . . . . . = 290 – 951 25 . . . . . . . . . . -3,992 3’36 a26 –3, 379 42. . . . . . . . . . - 128 - 4ss 497 627 26 . . . . . . . . . . –1, 923 –2, 779 43. . . . . . . . . . – 2s2 – 355 – 756 SW m 27 . . . . . . . . . . —5,870 —7, 320 44. . . . . . . . . . – 304 902 1,018 2s . . . . . . . . . . –2, 164 -3,732 45 . . . . . . . . . . – 555 – 603 404 404 29 . . . . . . . . . . –7, 109 –s, 7% 46 . . . . . . . . . . – 0S$ –lA 370 376 376 30 . . . . . . . . . . – us – 735 47 . . . . . . . . . . –050 –1, 334 1,955 2,210 31 . . . . . . . . . . – 561 – 0S5 48. . . . . . . . . . –1, Soo –2, 637 1,818 2,204 32 . . . . . . . . . . – 8S7 –1, 417 49. . . . . . . . . . –1, 912 –3: 038 3,879 4,432 33 . . . . . . . . . . – 35 a . . . . . . . . . . – 3S7 –1, 217 2E 3,217 4,039 34 . . . . . . . . . . – 254 I .
The results of the investiga~ions, as recorded in these tables, emphasize the great imporhmce of initial tension, the deleterious effects of which have too seldom been appreciated. In almost every instance the siresses under the worst probable distribution of initial tensions are greater than those which arise from the air load alone, either with or without-redundrmcies, III short, the stagger wires, as they are usually set up, are actually harmful and weaken the structure under most conditions of flight, whereas they should be an important elemenL of strength.
The initial tensions_ in the external drag wires are much more innocuous, although the values selectecI there S11OUM always be as small as are consistent with the rigidity of the stricture and with freedom from vibration when in flight. The stagger wires, being disposed in clirectly opposed pairs, can and should be so acljusted that there will be little or no unbalanced tension to affect the remainder of the truss. Even if this is done, however, the initial tensions should be kept small to ease the strain on the stagger wires themselves and on the interplane struts and drag struts or compression ribs which make up the parallelogram frames at-panel poin~s. If the alignment of the air plane’is carried out with a tensiometer the element of guesswork is definitely removed, the factors of safety in some important and badly stressed members are increased by from 25 per cent to 50 per cent, and the time required for rigging is increased very little, if at all, In fact, it is proboble that a crew which has had a little experience with a tensiometw can work quite as rapidly -with rLs without it~as the amount of trial and error required to bring the machine into true alignment is less than by the ordinary method.
In order that mechanics may have some reliable guide for use in rigging the designers of airplanes should draw up schedules of initial tensions to be used. The primary principle to be followed in drawing up such a schedule is that there should be no unbalanced tension in either of two diredy opposed members. In a rectangular fra,me this means tha~ the initial tension must be.equal. (This of course applies to the total tensions where there are two or more members in parallel. Where, for example, two flying wires oppose a single landing w-ire the initial tension in each flying wire should be just half that in the landing wire.) Where the frame is not rectangu- lar, but has two parallel sides, as in the stagger panels of an airplane with stagger or in the lift truss of a machine with interpla,ne struts sloping outwardly and with the same amount of dihedral in the upper and lower wings, the condition is that the diagonal wires should have In the case of a stagger panel, this means equal components perpendicular to the parallel sides.
that the tensions in the two stagger wires shoulcl be irmersely proportional to the sines of the angles~v~lic}l they make with the wing chords, so that the long clirtgona] has the larger iension.
In drawing up a tension scheclule the periods of vibration of all the wires should be high enough not to synchronize with the natural period of the engine, and should l)Q tipproximately the same throughout the structure. The fundamental frequency of a stretched wire can be T shown a to be equal to ~ where Z is the length of the wire, T the tensionj and m the 21 %’ ..~’extbook on Sound, J, II. Poynting & J. J. Thomson: p. s8.
!255 A2WALYSLS OF WIllG TRUSS STRESSES.
mass per unit length, -which is of course directly proportional &o the sectional area and so to the strength of the wire. The tension to gi~e a constant frequency must therefore be proportional to the ultimate strength and to the square of the le@h, _and it is necessary that very long wires be supported at some intermediate point, as the initial tension required to pre~ent vibration I& has been found by actual experiment that if this -were not done would be dangerously hQh.
-.
an initial tension of 220 pounds in the upper drag wire of a JN is enough to prevent vibration.
Since this wire carries an additional load of about 140 pounds when flying normally with a load factor of 1, the total reedanfi tem.ion for satisfactory results is 360 pounds, and this may be The flying and landing wires are taken as a basis for the determination of the other tensions.
substantially equal in length to the upper drag wire, but they have an intermediate point of support where they cross each other. The area of all these members are the same, and the resultant tension in the flying and landing wires must therefore be at Ieast 90 pounds (the With a load factor of 2, which is as high a value as is likely effective ]en@h being hslved).
to be maint ained st-eadiIy, the air load reduces the stress in the inner landing wires by about 630 p:mnds (the total air load on the wires in the inner bay being 1,S80 pounds, of which two-thirds is taken by an increase in the stress in the double flying wires, while the remaining third sho-ivs as a reduction in the landing w-ire tension), and the initial tensions therefore should be at least 720 pounds. The initial tension in each flying w-ire, as already noted, should be half this amount.
In the outer bay a tension of 390 pounds in the landing wires is sticient, as the air load eflect there is less. The length of the long stagger -wire is approximately two-tl@ds that of the upper drag -wire, and there is a center support where the two sta~~er wires cross. The area of the sta~~er wire is about half that of the external drag wire, so that the resultant tension for Nos. 6 ancl 7 in the eonspectus only needs to be one-eighteenth of that for No. 20, or 20 pounds. Under normal tom’ iions of flight (load factor of 2 or less) &he tension in the stagger wires is not changed mo -. than 30 pounds by the air load, and the initial tension thus does not need to Making some extm allowance to secure rigklity, 150 pounds for the long exceed 50 pc-:xls.
wire and 120 pounds for the short one appears ample, and tests in flight have shown it to be so.
The complete tension schedule for the JN is given below, and will serve as a guide in drawing up such schedule for ohher machines of similar type.
Iuitial Thitd tension tansion (iiclu!iin Member. , (ii<ludinz Member.
) w~ht of -~s).
-.. _ I Stagger wires,long . . . . . . . . . . . . . . . . . . . . . 155 , kertionb fl~g~w(wch] . . . . . . .
~ Sagwtim, tioti . . . . . . . . . . . . ... . . . . . . m Inuwrearflyingv nre.s(each) . . . . . . . ..i “ Front center sedion wires . . . . . . . . . . . . . .
Outer flying wires (each) . . . . . . . . . . . . ..l EQ Rearceutars wtionwirw . . . . . . . . . . . . . . . E W!an@.ng @a . . . . . . . . . . . . . . . . ..-l ~s Upper drag wire. -..-. -... -... - . . . . . . ..l H Lower drag wire. . . . . . . . . . . . . . . . . . . . . . .
,.:.. ‘,. , & d The pulls in the flying and landing wires in the inner bay are not exactly balanced because the vertieal components of the tensions in the external drag wires are balanced by motivation of the flying wire stresses.
PRACTICAL CONCLUSIONS AND SUMMARY.
The conclusions to be drawn from this work -will fimt be tabuIated and TviII then be exam- ined more in detail where they calI for such examination.
(~) The making of a least work analysis of a new design for at least one case is thoroughly justified. The labor of making such an analysis is not excessive and it gives an idea of the nature and magnitude of the true stresses which can not be obtained in any other way.
(ii) The wooden members may be omitted from com~ideration in the work equations without causing any serious error..
(iii) The effect of the stagger wires is unimportant when the load is ap_proximately equalIy distributed between the fl-ont and rear trusses. In diving the effect of the stagger wires is 54SS!3-21-17 ANNUAL REPORT NATIONAL ADVISORS COMMITTEE FOR .4ERONAUTICS.
very important, and greatly recluces the load on the lift trusses. The effect of the stagger wires depends in part on the wmmgernent of the. external drag wires.. If there is no external drag wire attached to the upper wing, and if the cemter section wires have as little forward inclination as they have on the JNj the stagger wires running upward from front to rear will be in termion at all times, transferring drag from the upper to the lower wing, and must he taken into account.
(iv) The tension in the external drag wires varies widely with the conditions of loading.
Only very rarely are both wires stressed at the same time, and most of the work now done by the two wires could be accomplished equally well by a single one.
(v) The initial tensions are almost always excessive, particularly in the stagger wires, and are sometimes so large as to be dangerous, especially as regards the compression ribs at, the lift truss panel points. The initial tension is sometimes so high that the total effect of the redund- ancies becomes harmful, whereas it should be distinctly beneficial to the total strength of the truss.
RECOMMENDA1’IONS.
I. OnIy one external drag wire should be used on each side of the plane of symmetry That one can be kept in tension nearly all the time, -whereas, as already noted, i~ is only rarely that the upper ancl ~oiver drag wires are in tension simultaneously. The structure. should of course be designed to fly normally (not to be stunted) without any external drag wires at cdL A single drag wire should be attached at the lower front spar, so that it will resist the downward and backward defection of the truss during a dive. If two external wires are used the second one should be attached either to the upper front or the upper rear spar. The first position is probably the more eil’ective in most instances, as the drag wire then reIie-iws the -mry heavy load on the front lift truss at large angles. The same result can be obtained without the use of a second drag wire by increasing the strength of the flying wires in the inner bay and attaching them to the fuselage a little forward of the wing spars, as has been done in several recent designs, Attachment of -t&edrag wire at the in order that they may resist the drag on the upper wing.
.— lower rear spar should not be emp~oyed.
11. The stagger wire which runs upward from front to rear carries a heavy load at times If a steel tube, with no opposing and may well be made stronger than the other diagonal.
member, is used for stagger bracing it should run upward from front to rear. If there _is no drag wire attached to the upper wing, such a tube need not be designed to carry a compressive load of more than one-eighth the weight of the airplcme, but it should be capable of sustaining a tension equal in magfiitude to the iotal weight of the machine. If picture-frame struts am used, and they are highly recommendecl, they should be clesigned to carry from five to eight times as large a compressive load in the direction of the long diagonal (for a machine with positive stagger) as in the direction of the other diagonal.
III. Airplanes should be rigged, whenever possible, by means of a knsiometer and in Detailed instruc- accordance with a schedule of initial tensions to be provicled by the designer.
tions for drawing up such a schedule have already been given. In partimdar, the tensions in the stagger wires should be far less than has been the common practice, and opposing members One great adyantage of the picture-frame strut-is Lhat it should exactly balance each other.
eliminates all danger of excessive initial tension.