Bruhn - Analysis And Design Of Flight Vehicles Structures Episode 2 Part 9 potx

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A22, 2

Fig A221

The sheet-stringer panel may, in general, contain a large number of longitudinal elements (stringers) The labor involved in treating this multi-element structure in detail is prohibitive, and thus an appropriate idealization must be made First, it is likely that the panel may be considered to be symmetric about a longitudinal axis, so that only the falf-panel need be handled Second, the complex, multi-stringer structure is replaced by one naving but three stringers As indicated in Pig A22.2, these stringers are:

#1, a substitute stringer having for {ts area all the effective area of the fully continuous members to one side of the "combing stringer" (the stringer bordering the cutout) and placed at the centroid of the area of material for which it substitutes The stress which this stringer develops is then the average stress for the material it replaces

#2, the combing stringer, being simply the main continuous stringer bordering the cutout

#3, another substitute stringer, this one replacing all of the effective material made discontinuous by the cutout It 1s located at the centroid of the material it replaces, and its stress is the average stress for this same material.*

The sheet thicknesses used are the same as those of the actual structure."

* An alternate idealization, in which stringers #1 and #3 are

located along the lines AB and CD, respectively (Fig A22 2),

was used in Reference (2) for a box beam loaded in torsion ** When the longitudinal members themselves contribute to the shear stiffness of the cover (as is the case for "hat" section stringers riveted to the skin so as to farm small closed cross sections}, an effective thickness must be used This point is discussed in Reference (3) In this source, however, the increase in shear stiffness is accounted for, not by increasing skin thickness, but by decreasing the panei width -

ap equivalent procedure

ANALYSIS OF SPECIAL WING PROBLEMS

Fig A22.2 Idealization of the half panel by

use of substitute stringers

Fiz A22.3 gives the geometry of the idealized panel A,= 703 in? TT Lo A,= 212 in = in? +4 | Ag= 1.045 in? Loi a + AR= 0.25 in ub! t, = 0.0331 in tw ' t, = 0,0332 in ° bạ 5.86 in ae A, Ag "ty tạ b,=7.56 in Fig A22.3 L =15.0 in SOLUTION:

Fig A22.4 1s an exploded view of the halfr- panel showing the placement and numbering of the internal generalized forces (Art A7.9, Chapter A7) and the external loading Note that the applied axial stresses were assumed to be constant chordwise, giving stringer leads poro- portional to the stringer areas; their sum is Py, one of external loads

The applied edge shear flows, coming from the spar web, were assumed constant spanwise, as from a constant shear load Other load distributions may be handled by allowing these applied shears to vary from panel to panel For very extreme load variations additional transverse members could be inserted to create more spanwise panels allowing a better fit to the spar shear variation The applied shear flows were considered as the other external load and designated P,

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ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES = 389 (P,+3P, L) = 108 (P, +3P, L) meee g7 qa SWAT = Ấn, aye 108P, 4, £.538 i Fig A22.4

Experience has shown that for symmetric panels symmetrically loaded it is satisfactory to consider transverse members to be rigid in their own planes (4) Thus, in this problem, member flexibilities for forces 4,4, I,» 4, and GQ, may be taken to be zero In the actual NACA test specimen with which results are to be compared, those transverse members bordering the cutout appeared to have been heavily reinforced (to an extent unknown to the writer) Hence it is logical to take their stiffnesses

as great

#ember flexibility coefficients were collected in matrix form as below using the formulas of Chapter Av’ Note that the coeffictents for subscripts 5, 6, 7 and 8 were set equal to zero (rigid transverse members)

o<\j, MEMBER FLEXIBILITY COEFFICIENTS {E 21) 6 7 a 9

A22.3 The hal? panel was twice indeterminate Member loads q, and q, were selected as redundants With these set equal to zero, successive apolications of loads P, and P, were madé Also, successive applications of unit values of q, and q, were made The results: it 1 2 6 8 1 | 4.7355] ,516 |~L.68 |~ 840 2 | 9.510] ,516 |~ 840|~1,68 3 |-19.76 |~l.Z18 | 1,68 | „840 4 |-39.51 |-1.316 | ,840| 1,68 5 9 |- 26a | 1.00 | 0 6 0 9 1.00 | 0 7 |-12.09 |- 268 [ o | 1.00 8 0 9 9 | 1.00 [Etni=rr] 9 |-1.517J- 0450| 112| 056 10 |-1.a17| 0 |- 9086] ,056 11 711| 0450|~ 066|~ 112 12 0 %88| 0 0 15 |- 1.599|- 0558| O 9 14 ọ 359 | O 9 15 9 wos | 0 9 16 ọ 533 | 0 9 17 | ies | 359 | 0 9 18 | 4/86 | 1Q8] 0 ọ 19 | 25.98 | sso | 0 ọ (Voids denote zeros) 10 11 12 18 14 15 18 1? 18 18

The "off diagonal" values have negative sign because the sense of those internal generalized forces having subscripts (14), (15), {17) and (18) was taken opposite to that used in the derivation in Art A7.10 A change in sense requires a change in sign in off-diagonal coefficients only.)

32

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A22.4

The following matrices were formed: Per eq (17) of Chapter A8: [=m.]* Per eq (18) of Chapter A6: = 293.5 [-<rs | 7 soe | The inverse cf? this last was found; E= ¬ =z10 Ẻ

Finally, per eq (23) of Chapter A8, the unit load stress distribution was, ~5206 6633 262.8 -262.8 392.1 293.5 5.802 4.343 4,343 5,802 N 1 2 1||-10.95 | -.65 2|Ï-14.34 | -.65 ~ SiÌ- 4.08 | -.351 4||-11.66 | -.51 SỈ 1.40 | 118 6Ï 1.40 | 385 7| 5.79 | 115 ai] 15.88 | 385 [em ] = giÌ- 270| ,019 10ÌÏ= 5l0[ 0 11lÌ~ 1.18 | =.018 || 9 0555 13 |} - 1.60 | -.0355 uo 559 isto -108 1| 0 ec) 17|| 16.16 | 359 15|[ +.88 | 1Q8 19ÌÌ 23.s8 | 559

The above analytical results are compared with NACA test data (1) in Fig A22.5 for the

loading P,= 1, P, = 0 Agreement ts seen to be good

EFFECT OF RIB FLEXIBILITY:

To investigate the influence of rib flexibility, the problem was reworked assuming aluminum rib caps, of constant area A = 25 in.4 as the transverse members bordering the cutout The appropriate member flexibility coefficients ANALYSIS OF SPECIAL WING PROBLEMS CALCULATED Fig A22.5 MEASURED

Comparison between calculated and measured stresses (psi) on the half-panel,

inserted in thee<,4 matrix (in place of the

zeroes used above) were J 1 18.03] 5.04 [><] = 5.04 | 10,08 5.04 10.08 5 8 ? 18.03 8 5.04

The problem was solved retracing the same

steps as before, but using the mod171ed%€ ¿|

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ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES Comparison of this result with the pre-

vious one for the rigid ribs reveals that the most important effect of rib flexibility was to increase the concentration of stresses in the combing stringer bordering the cutout It should be noted, however, that for this symmetric panel, the use of a very flexible rib as compared with a rigid rib led to stress increases of the order of only 10%in the combing stringer Thus, the "rule of thumb" that transverse flexibilities may be neglected in sum- metric panels is re-affirmed

A22.3 Shear Lag Analysis of Box Beams

"The bending stresses in box beams do not always conform very closely to the predictions of the engineering theory of bending The deviations from the theory are caused chiefly by the shear deformations in the cover of the box that constitutes the flange of the beam The problem of analyzing these deviations from

the engineering theory of bending has become known as the shear lag problem, a term that is convenient though not:very descriptive." (3}

Fig A22.6 illustrates the basic problem The beam cover sheet is loaded along the edges by shear flows from the spar webs These shear flows are resisted by axial forces developed in the longitudinal members (spar caps and stringers) According to elementary consider-~ ations, the stringer stresses should de uniform chordwise at any given beam station ("elementary theory" in the figure) Actually, the

central stringers tend to "lag behind" the others in picking up the load because the Ì ly : 1 4 : iA a y | Z1 ZZ / | x⁄ l1 Z V ly ! ZZ i Z i i Z iA ` Elementary Theory — —-Actual Stress Fig A22.6

intermediate sheet, which transfers the loads in from the edges, is not perfectly rigid in

A22.5 shear The action may be comprehended readily by visualizing an extreme case: a large degree of "lag" would occur if the load transferring skin were made of a highly flexible material such as a plastic sheet or even rubber In such a case the inside stringers would be out of

action almost entirely! With the inside stringer

stresses lagging, the outside stringers and spar caps must carry an over-stress to maintain equilibrium ("actual” in the figure)

Fig A22.7 shows the beam analyzed herein - 636 ~+4s@8" Ệ ‘An 8494 A=.378 A= 349T Fig A22.7

The beam is an idealization of one tested by the NACA and reported in reference (3) Note that the beam has no lower cover sheet and that

it is symetric about a vertical axis Trans-

verse bulkheads are located at stations 12", 24", 26" and 48" from the root

The actual beam specimen had three more stringers than shown in the tdealized structure, these being located one each midway between the pairs of longitudinals shown on the beam cover of Fig A22.7 In the idealizing process, these extra stringer areas were divided equally between adjacent longitudinals The stringer areas shown are the effective areas, with those in the top cover tapering linearly from root to tip All skin was considered effective in carrying direct stresses

Some detailed discussions of the techniques of idealization of practical beams are given in

references (5) and (5a)

SOLUTION:

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A22.6

the structure for a single transverse (vertical) tip load symmetrically placed In that case, because of symmetry, it was necessary to treat only one-half of the structure In addition, no shear flows could appear in the middle

panels Further, {t {s known that the influence of rib flexibility on shear lag is slight for symmetric systems, so that the ribs were considered rigid in their own planes; hence no generalized forces were needed on the ribs to describe their strain energies

Fig Az2.8 shows the placement and number~ ing of the generalized forces on the half-beam a Ne SN ng KG XS ti ` an: SN 4a oO DRESS cv & be SRS Fig A22.8 Choice of generalized forces for shear lag problem

Member flexibility coefficients were computed with the formulas of Chapter A7 and arranged in a matrix

The shear flows q,, 4,, 4,, and q,, were

Selected as redundants Setting these equal

to zero, the stress distribution due to 4 one- half pound load at the tip ( 3 unit load

divided equally between beam halves) was readily computed, f „07692 ~.8230 +9230 9 „07692 9 , ~1,8161 1,8461 9 „07692 li ~2.769 2.769 9 „07692 9 -5.692 3.692 9 5e] - +2

Next, the unit redundant stress distribution was computed Fig A&2.9 illustrates a typical calculation, showing the stresses in the tip bay for q,=1, 4,79,,74,,70

o<jj, MEMBER FLEXIBILITY COEFFICIENTS (E = 1)

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ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES A22,7 % , Constant on in ue í = (7 L2 ` to Root rr -1260 ° ~ 460.7 40,210 16,810 9,556 3,041 Sek ack ` AeA VỆ aaa C_ = | 16810 32,190 9,535 3,041 T8 9,526 9,536 25,030 3,041 3,041 3,041 3,041 18,650 XÀ The inverse of the latter was formed: End Rib aN 3.285 -1.505 - 656 = 163 ale -6|~+1.505 4.219 -1.000 - 279 [=s"] #1 | 684 -1.000 4.6887 - 495 Fig A22.9 Application of a self-equilibrating unit ~ 183 - 279 - 495 5.519 redundant stress q, The complete redundant stresses were:* r 1 7 12 17 (ol ø[ sa | oi oi [ olelololololiolo ‘ fo H ny oilololololol|olololoclolio Ít | |b|tr| e Bb (mi ml || ¬ pe] Bl Bl olololb 1 ưoll|ojolocololololocoleloleielocolsel|selolso -12 | -12 | -15 20 12 12 12 12 i Bb tị

The following matrix products were formed

(per eqs (17), (18) of Chapter A8):

* This is an obvious place in which to use combinations of redundants to decrease the structural coupling (reference Chapter A8, pp A8.29, 30) (Recommended as an exercise

for the student.)

and, finally, per eq (23) of? Chapter A8, z „07692 „03453 ~ „9880 „5086 4144 „07692 „08126 ~1.846 @ = 1,085 im 7207 _ 07692 02401 -2.7691 1.690 1.079 07692 „01008 ~5.692 2.498 1.199

Fig A22.10 snows the above computed stresses and those reported by the NACA as ob-

Agreement Is seen to be quite

tained by test good

MEASURED

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A22.8 ANALYSIS OF

A22.4 Stress Analysis of a Box Beam With a Cutout

In Article Ac2.2, ome technique was employed for computing the stresses around a4 cutout In that analysis the effect of the cutout was presumed to have been localized about the cutout region; consequently, the problem was treated by isolating the affected panel Quite often, when the cutout is placed well inboard on the wing, its influence on the root stresses is appreciable Therefore, it is desirable to be able to consider the overall problem of the box beam with a cutout for such cases

"The most convenient and the most rapid method of analyzing structures with cutouts is the indirect, or inverse, method The analysis by the indirect method is made in two steps First, the structure is analyzed for the basic condition that exists before the cutout is made The results of this basic analysis are used to calculate the internal forces that exist along the boundary of the proposed cutout External forces equal and opposite to these internal forces are then introduced; these external forces reduce the stresses to zero along the boundary of the proposed cutout, and consequently the cutout can now be made without disturbing the stresses." (3)*

It is desired to modify the calculation of the previous article (A22.3, "Shear Lag

Analysis") to allow for the presence of two cutouts symmetrically placed Panel "q,." was removed while the single tip load remained Since the unit remained symmetric, the data from the previous analysis, in which the transverse rib stiffnesses were taken to be

infinitely great, should still yield satisfactory results

SOLUTION:

The calculation was accomplished in three steps:

1) The stress distribution was found in the "basic structure” (no cutout) This work was carried out in Article A22.3 where it was

found that q,, = 02401 lbs./inch

2) A stress distribution was found for an "applied load" of q,, = 1 Such a loading

has zero external resultant

3) 02401 times this last stress distribution was subtracted from the ‘first

* The procedure described here is quite generaily useful for

studies of the effect of removing one or more members; such might be required for an analysis of the effecta of

structural damage

SPECIAL WING PROBLEMS Tre calculation for now be carried out Dy co

flow, q,, 2 1, applied whi 1

are zero Under this loadi oncition the relative displacements at redundant cuts 2, 7 and 17 are equal to -<, ,,, ,,,, amd “.,,., respectively, where these coerfZicients have been computed previously in article Ag2.3

other loads

To restore continuity (reduce the relative displacements to zero) three redundant forces are applied, one each, at the cuts 2, 7 and 17 The appropriate equations specifying continuity are —_= of “¿a2 897 ayrr Qe ayaa MS ea Maye Meare a, F.C ua AT v7a+ ®vxyvz Sự; TS, ca

Note that these equations say sin2ly that the deflections at the redundant cuts due to the (unknown) redundant forces must 5e equal and opposite to the deflections due to q,, 51

All coefficients in the above equation were computed in Art A2Z.3 Specifically,

40,210 18,810 3,041 a, 9,536

16,810 32,160 3,041 a, = - 49,526

3,041 3,041 18,650 LÊ 5,041

The above matrix 1s the [<rs] of Art A22.3, with the "q,, row" and "q, column” removed

The equations were solved** to give the values of the redundants for a unit applied load, q,,21, as q, - 1399 a, ~ 2154 {22} *\ al 1.00 dis - 1055} a, =1

The q,, force is included in the above for

later convenience Then the complete stress distribution, due to applying a unit shear rlow Gua = 1, was

** See Appendix for a method of "extracting" the inverse of

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ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES {or,.0} 2 [err] {sr} ~ 1899 9 1.679 -1.879 9 - 2134 0 + 4,240 = ~4.240 \ 9 1.00 9 ~7.760 7.760 9 - 1055 9 ~6,494 6.494

( [si ] teken from Art A22.)

To obtain the stresses in the loaded beam with the cutout, 0.02401 u.nes the above stresses were subtracted from the "basic" stress distribution of Art A22.3, as previously explained „07692 „05799 - 9250 „4685 4547 207692 „05648 ~1.8461 „9536 „8925 „07692 9 ~8.7691 1.8765 ,8927 07692 01258 -5.692 2.649 1,07 Suy) sụn) ~.02401(G }s mo THÍ S1C 1ixa

Note that q,, 1s now zero and the cutout panel may be "lifted out”,

In the case of a structure under 4 of external loadings (m # 1, 2, 3-~-), more general equation, corresponding to above, is [%: = [21a] “1 L Sa @ cuTOUT ™ pas ic oua variety the the A22.9 where the row matrix [Siam ig simply the "12 row" of (1m |sasrc

COMPARISON WITH TEST DATA:

Reference 3 reports test data for the case analyzed, the stringer stresses being plotted >delow in Fig Ace.11

In the actual test specimen a stringer passing through the cutout was severed, it having zero stress at stations l2 and 24, therefore However, during the idealization process discussed in Art A22.3 (for the beam without cutout) the area for this stringer was placed partly with the combing stringer and partly with the spar cap In the same way, some effective sheet from the midportion of the panel, now made discontinuous by the cutout, was added also to the spar cap and combing

stringer

It follows then that the full idealized areas of the combing stringer and spar cap

Should not be used in figuring the stresses at

stations 12 and 24 (produced by forces dạ; Gucs Guar Tisle With these areas reduced by the appropriate subtractions, the stresses were computed and are plotted in Fig A22.11 Agreement with test data is seen to be quite satisfactory

MEASURED

Fig A22.11 Comparison between calculated and measured

stresses (psi) in a box beam with cutouts

A22.5 Analysis of a Swept Box Beam

"Experimental investigations of swept box beams nave shown that the stresses and distortions in a swept wing can oe appreciably differ-

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A22 10

shear in the rear spar near the fuselage With regard to distortions, the effect of sweep is to produce some twist under loads that would produce only bending of an unswept wing and Some bending under loads that would produce only twist of an unswept wing." (6)

In the following example a swept box beam 1s analyzed by the matrix methods of Chapter A8 and, in particular, by the specific techniques of reference (7) The method accounts for the interaction between the swept cover panels and the longitudinal members It is this action that is responsible for the distinctive structural characteristics of the swept box beam

Again, we emphasize that the method used here is strictly applicable to thin-skinned wings of beam-like proportions only Consider-

ing the wide variety of structural layouts which may be employed in swept wing configurations, a comprehensive treatment cannot be given here An excellent review of methods better adapted to thick-skinned construction and to "plate-like" (very thin, wide) wings, may be

found in reference (8) One method of analyzing such wings is zZiven in Chapter A23,

THE STRUCTURE:

The structure shown is fig A22,.12 ts an idealization of the NACA test beam of references (6) and (9), in which a single substitute stringer has been employed along the cover sheet to allow for the antitipated shear lag effect The figure shows only one-nalf of the complete unit, which was built symmetrically about the axis corresponding to the longitudina? axis of the airplane

Only tip loads were to be applied (at points A and B) The outer section of the beam was assumed to carry stresses which could be calculated reasonably well by the engineering theory of bending (E.T.B.} For this purpose it was judged satisfactory to consider the outer 66" of the deam as a Single bay (A-B+D-cC), If loads were to have been applied inboard of the tip, it would nave been necessary te con~ Sider additional bay divisions between 1-3 and C-D (that is, insert additional ribs at staticns 4 i gr An 538 lo | Section C-C 050 -050 | (Steel) “pets 125 66” | ee A=l.3743 ,A=l.121 | 078 4- section A-A a | * — ae

‡ te L tị J | } Em TF A=1.966 A=1, 430 pas jn 8 yal Ni %4

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ANALYSIS AND DESIGN OF PLIGHT VEHICLE STRUCTURES of load application) Rib C-D was located at

one of the actual rib locations in the NACA test specimen and was assumed rigid in its own plane

The choice of bay C-D-F-E as a single bay was somewhat arbitrary Por improved accuracy, additional ribs inboard of C-D could nave been used, Note that any ribs placed inboard of point F will produce triangular skin paneis in the cover sheets Examples of treatments for such panels may be found in references (9),

(10) and (11)

Rib E-F was considered flexible in its own plane, it being known that the flexibility of a rib is important at a locatiun where a structure changes direction.* Note that this rib was made of steel in the test specimen

Effective areas of longitudinals as shown in Pig A22.12 were computed by considering all of the skin to be effective The spar cap areas are equal to the sum of the areas of the angle member at the cap location, plus one- nal? of the effective area of material between the cap and the substitute stringer (this area

includes several stringers as well as skin} plus ome-sixth of the attached spar web area** The substitute stringer area was collected in

like manner from the half-panels to either side The method used in calculating the

effective areas of the rib caps (E-F) is given in detail in refarence (9), from which the value used here was taken The ”carry-through bay" cover sheet thickness {s equal to that~ used on the specimen (.050") plus a weighted tnerease to allow for the presence of splice plates along the plane of symmetry (see refer-

ence 9)

INTERNAL GENERALIZED FORCES:

Fig A@2.13 shows the choice and numbering of the generalized forces

The beam was rigidly supported at points &, F and at the two corresponding points on the other beam half These might correspond to the fuselage ring attach points in an airplane The vertical end caps on rib E-F were considered, rigid axially, ¢o that no flexibility co-

efficients were associated with the reactions Ga, and dag Flexibility of these members

* The effect of neglecting this rib’s flexibility is demon-

strated later in this exampie

** The factor of 1/6 is used so that the effective area con-

tribution of the web results in a structure having the same moment of inertia about a horizontal axis as the original Some of the problems of idealization are discussed in reference (2}, p 16

A22,i1 affects total deflections only and can be omitted in a stress analyses where deflections are not sought

Since only symmetric loadings were considered in this analysis no shear was trans- mitted by the carry-through bay and hence no shear flows were shown in that portion

Sets of additional axial forces (q,, through q,,) were applied to the ends of the flanges and stringers adjacent to the obliquely cut ends of the cover sheet panels in bay C-D-F-E These forces are necessary to account for the interaction between the swept covers and the longitudinals As shown in Fig Ae2.14, the pure shear flow on the oblique edge is obtained by superposing onto the panel a zero=— resultant system consisting of a uniform tensile stress of intensity 2q plus a pair of concen- trated balancing loads The balancing loads must be contributed by the bordering longitudinals and hence react on these as tensile loads

(Flg A22.14c) The balancing loads applied to the stringers are shown dashed Since they are internal forces within the bay and are not to be entered into the equilibrium equations for the structure Pegh P Pa “EVE ZT sro}! = = (a) (b) oO

Fig A22.14 Showing how the uniform shear stress on an oblique panei end (b) is created by

superposition of a uniform tensile stress

plus two balancing forces {a} The balancing forces react as tensile loads on

the bordering longitudinals (c)

From an energy viewpoint, these dashed forces account for the additional strain energy stored by the axial components of shear flows in the non-rectangular panels This energy is stored in the cover panels themselves (and 1s accounted for in this manner since the longitudinals contain the effective area from the cover sheet*** ) and in the longitudinals which react against these components

"Dashed loads" are applied to the longt- tudinals adjacent to any obliquely cut panel end Similar dashed loads would be applied to

*** This much of the energy could be accounted for in another

fashion by modifying the member flexibility coefficient for the sheet panel See Reference 12, where this was done,

However, that reference incorrectly neglects the additional

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A22 12

the outboard ends of the panels in bay C-D-F-E if they too were cut obliquely Such oanel configurations arise often in swept wing construction having ribs parallel to the air~ Stream Formulae for more general quadrilateral panels are given in Reference 7

THE STRESS DISTRIBUTIONS

For the symmetric loadings considered here the structure was indeterminate only two times since the outer bay was assumed to be determinate oy the E.T.B

Stresses in outer bay by 5.T,B.: Flange stresses at rib C-D (for both P, = 1 and P, = 1) M = 661% I = 88.57 in.*; ¢ = 3,5" fh at = 2.608 psi Therefore, ds = 4, 2 2.608x1.121 = 2,924 lbs d, = 2.608x1.373 = 3.581 lbs

For a unit transverse load at the shear center (midpoint, because of symmetry) dạ =ốt Sen 02718 195 2.924 gp Wb = 02713 « 755 = 07143 ibs in

The unit load was shifted 15" to either side by application of a torque, T = 15 in.lbs The uniform shear flow superposed was

T lễ

3“zr°#x BIO = ,08571 =

Finally, superposition gave the stresses for the outer bay as ` 1 2 1 1071 08572 2 106284 | - 00858 [ | 3 „00858 ~ 06284 4 03572 +1071 5 2.924 2.924 6 3.581 3.581 7 2.924 2.924 Stresses in inner bays: 8.12, According to the discussion of art A stribution,

Chapter A8, the determinate stress di

Eta , may de any stress distribution in equili- ibrium with the applied loads, and preferably one close to the final true stress distribution Ths magnitude of the redundant forces is

duced by use of 4 Satisfactory estimate of true stresses, ©

re-

the

The stresses in the two inner bays were determined for both gin and gyy simultaneously Since this inner portion of the structure is two times indeterminate we can estimate two loads For this purpose the two flange loads Qi, and q,, were written as

Gia = dia + Ga

3 2 41, + Q,

where the (single) primed values are approxi-

mate values determined by the 5.T.B and the double primed values are the unknown corrections

(the redundants) Using My/I at stations 66" and 118" from the tip gave*

Sa de

3.228 PL + 5.228 P, 3.899 R + 3.899 P,

The equilibrium equations for the elements of the structure were written next by suming forces and moments Joint F Gas + Vag 70 des + Qa, = 1.414 Gy, = 5.513 P, + 5.513 P, + 1.414 qt, Joint £ Waa * Tao = 0 Gag ~ Gao F 1-414 Oy, = 7.392 P+ 7.392 PL + 1.414 g™, mM about

daa + dee + Uy, 7 11.918 PL + 8.888 P,

* This is a rather crude way to estimate these loads and is

used here only for simplicity The analyst is generally better advised to exercise a little more ingenuity in making these estimates, even to the extent of being guided by other swept wing solutions

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ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES A22.13 ZG 1 9 9 0¬ 9 0 de : 1 -.715 0 90 9q 00 ae ° a 9 9 212 0 21.2 0 1 Cao = 9 Oo 1.682 1 0 00 ai, $= c 0 9 0 1 1 00 qua ca O -21.2 0 0 -21.2 10 Gar

Shear Flows around the non- 9 9 9 9 9-1 1 đạa

rectangular panels (Check

by summing moments about E, Gand F.) ~ 1405 ~ ,2111 9 0 „044261 „04451 01925 9 Rr cap Be 2.756 2.756 9.707 PL Q,- «71154, = -O4431 P, + 04431P, + z| 04432 04432 o ,04545|/ T2 " .01923 a", +2831 „3540 9 ° Sa at Rib Vertical at = 3.696 ~ä.696 —.707 9 Le 5.466 2.436 2707 «707 Ge 4,,2- -1403 B - 211 P,

cap BG After inverting the matrix of coefficients

= on the left hand side of this last equation,

« and multiplying through thereby, the stresses

~21.2q, ~ 21.20, ~Gaot Gar * 0 were obtained as : Ys Joint G đa „10Q6 ‹0295_ 00676 -.00676 707 707 Ga ~ dae = 0 - = qe .0791 - 0208 ~.01753 -.00950 +707 Gis ~ das + Gan = 9 d,,| | -0013 - 0411 ,00401 0250 | [Pa cap GF aii}=| 0422 1135 ~.00675 ,00675 ?› —— — 7 ar a Que + Tang + 04717 (Gan Gag) 29 dis 2409 2409 00676 00676 a Geil | 3-087 964 -.0855 -.3444 14 1 Rib Vertical atT Gaal |-2.379 -1.478 -.2284 3625 q¿, + Q,„ 7 -2831 PL + 3540 P,

Cap DF The complete determinate and unit-

——— redundant stress distributions, using the re-

q,, + 1.682 q,, = 04452 (BL + P,) sults up to this point, were therefore:

+ 04545 qi,

The first five of the above equations were readily solved by substitution, ylelding:

Vas 7 Vag = 2-756 P, + 2.756 P, + 707 af, “Vn = Vag = 3-696 PL + 3.696 P, + 707 at, dg, Z 5.466 P, + 2.486 P„ - 707 Q1, : - 707 at, Also, from equilibrium of joint G: a,5 = 7-729 P, + 3.444 PL - a2, - Uy

The remaining seven equations were

Trang 13

A22.14 Oey ot 2 12" 1a" 1 | 1072 | 0572| s0 9 2 | 06284|- 00858] 0 ọ 3 | 00838|- 062844 0 0 4 | 0572| 101 | 9 0 | 2.924 | 2.924 0 ọ 6 [3.581 | 3.581 ọ 9 7 | 2.924 | 2.924 0 0 8 | 1006 | 0296 | 00676]~ 00676 9 | 0781 |~ 0208 |= 01783|~ 00850 10 | 0018 |~ 0411 | 00401| 0230 11 | 0422 | 1135 |= 00675| 00675 | 12 |5.228 | 6.228 | 1,0 9 fein isin) 15 | 7.729 | 3.444 |-1.0 |-1.0 14 | 5.899 | 5.899 9 1.0 ˆ 185 | 989 | 260 | Ø191 | 118Ơ 16 1.384 | 360 | 3068 | 1662 17 | 0146 | 460 |= 0449 |= 2576 18 .0244 | 773 ]- 0784 |- 4624 19 | 2409 | 2409 | 00676]~ 00676 20 4.696 |-3.696 |~ 707 3 21 |s.cs7 | 964 |~ 9355 |_ 3442 22 |-2.379 |-1.473 |- 2284 | 3625 23 | 2.756 | 2.756 0 „707 24 | 3.696 | 3.696 | 707 0 25 | 5.466 | 2.456 |- 707 |- 707 26 | 2.756 | 2.756 Q 70?

DASHED LOAD CALCULATIONS:

In the above matrix, loads q.,, die Ur and q,, were obtained from q, and q,, following formulas given in reference 7 for general quadrilateral panels The equations applicable to a parallel-sided panel are: qa Porgrd x2 Po € + Pp^g+~a qLx2 —— a fa + Balancing Load Pp at (= Dashed Load)

Forpaulae from Ref 7

For the panel CMGH; c = 52.3", d = 73.5", cea = 125.8", Ls 15" Hence ANALYSIS OF SPECIAL WING PROBLEMS dis = "Pp" = 2 x peer x 18 x (+h ay = 712.5 ae = -.989 P, + 260 P, + 2191 gq", + 1168 al, = 73.5 =< = Qe = "Po" = 2 x TREE x 1S x (-) a, = -17.5 a, = -1.384 P, + 560 P, + 5068 q™, + 1662 0%, Simtlarly, for the panel HGrD; diz = -.0146 P, + ,460 P„ -,0449 q1, - 2576 Gig = ~-0244 PL + 773 P, -.0754 gt, ¬ 4624 q?,

The following matrix products were formed:

Trang 14

ANALYSIS AND DESIGN OF F

MATRIX OF MEMBER FLEXIBILITY CO!

COMPARISON WITH TEST DATA:

Station (0) ~

CALCULATED MEASURED

Fig A22,15 Comparison Between Calculated and

Measured Stresses (psi) in a Swept

Box Beam

Fig A22,15 shows a comparison between the calculated stringer stresses and those measured by the NACA as reported in Reference 6 The stresses shown are for a unit tip load, centrally placed (P, = P, = 1/2 lb.)

Considering the limitations on the analysis the agreement is generally satisfactory Thus, the discrepancy in the My/I stresses at station 65 may be attributed to a lack of precise knowledge of the test parameters The calculated stresses in the leading edge spar between stations 65 and 118 cannot duplicate the experimental variation since only a single bay was employed in this region in the idealization The fact that the calculated root stresses run consistently above the test values is airficult to explain Inasmuch as the calculated stresses! Satisfy equilibrium, the test values, all being lower, would seem to defy this fundamental

LIGHT VEHICLE STRUCTURES

(E = 1} (Formulas from Chapter A7 and Reference 7)

A22, 15

13 | 14 | 1S | 16 | t7 | 18 |19| 20 | 21 | 22 | 23 | 2z | z5 | 26

requirement More details concerning the test- ing techniques and method of data presentation would probably resolve this conflict Both test and calculated values clearly exhibit the

characteristic build-up of stresses in the rear spar of a swept wing

RIB E-G-F RIGID:

AS a4 matter of interest, it was decided to investigate the effect upon stresses when rib E-G-F is taken to be rigid Such a calculation ts readily achieved by putting the member flexibility coefficients for the rib equal to

zero,

Thus, in the natrix[=,; | those coeffictents with subscripts 19, 20, 21, 22 and 23 were set equal to zero and the complete calculation was repeated

Trang 15

A22.16

Considering that rib EGF was relatively rigid to begin with - being made of heavy gage steel - it may be seen that neglect of the flexibility of a corresponding ali-aluminum rib could lead to serious errors

REFERENCES

(1) Kuhn, P., Duderg, J E., and Diskin, J H., Stresses Around Rectangular Cut-Outs in Skin~Stringer Panels Under Axial Load ~ II,

NACA WR L368 TARR 2902), Oct 1943

{2) Rosecrans, R., A Method for Calculating Stresses in Torsion-Box Covers with Cut-

outs, NACA TN 2290, Feb 1951

(3) Kuhn, P., and Chiarito, P T., Shear Lag in Box Beams: Methods of Analysis id Experimental Investigations, NACA TR 739, 1942

(4) Hoff, N J., and Libby, P A., Recommenda-

tions for Numerical Solutions of Reinforced Panel and ruselage-Ring Problems, NACA TR

934, 1549

(5) Kuhn, P., Approximate Stress Analysis of Multi-Stringer Beams with Shear Deformation

of the Flanges, NACA TR 636, 1936

(Sa) Kuhn, P., Deformation Analysis of Wing Structures, NACA TN 1361, July 1947 (6) Zender, G., and Libove, C., Stress and

Distortion Measurements in a 45° Swept Box Beam Subjected to Bending and Torsion, NACA TN 1525 +

(7) Wehle, L B., and Lansing, W., A Method for Reducing the Analysis of Complex

ructures to a Routine Proce Aero Sci., 19, Oct 1952

(3) Williams, M L., A Review of Certain

Analysis Methods for Swept Wing Structures,

Journ of Aero Sci., 19, p 615, S52 (9) Helden?els, R., Zender, G., and Libove, C.,

Stress and Distortion Analysis of a Swept Box Beam Having Buixheads Perperncicular to the Spars, NACA TN 2232 ~

(10) Bisplinghoff, R., and Lane, 4., An In- vestigation of Deformations and Stresses in Sweptback and Tapered Wings with Dis- continuities, Mass, Inst of Tech Reot., July, 1949

(See also, Journ of Aero Sci 18, p 705,

1951)

(11) Denke, , The Matric Solution of Certain Non-Linear Problems in Structural Analysis,

Journ of Aero Sci., 28, 1956

(12) Levy, S., Computation of Influence Co- efficients for Aircrait Structures with

Discontinuities and Sweepback, Journ of

Kero Sel., 14, Oct Lo47

(13) Islinger, J 8., Stress Analysis and Stress Measurements for a Swept Back Wing Having “ibs Parallel to the Airstream,

McDonnell Aircrart Corp Report 1127, April

Trang 16

CHAPTER A23

STRUCTURAL ANALYSIS OF A DELTA WING ANALYSIS BY THE “METHOD OF DISPLACEMENTS”

ALFRED F SCHMITT A23,1 Introduction

The purpose of this chapter is two-fold: first, to present a discussion and an example of techniques for handling the structural analysis of highly redundant low-aspect ratio wings (typified by the delta wing) and second, to illustrate the "Displacement Method” of

structural analysis (1)*, a method fundamentally}

different from those of Least Work or Dummy- Unit Loads

The very low aspect ratio, thin wing is a structural configuration of relatively recent origin It is a built-up structure of ribs, Spars and a cover sheet but yet is so thin and so highly redundant that its structural

characteristics are actually closer to those of a tapered flat plate than to a beam Ina beam the primary stresses are longitudinal; indeed, one of the basic assumptions of beam theory is that transverse stresses are negligible and that cross sections remain undeformed For the very low aspect ratio wing however, chordwise stresses and deformations are of great im- portance The degree of redundancy of these low aspect ratio wings is very high because of the multiple paths by which a load may be carried over the gridwork of spars and ribs

In the wing beam problem examples of Chapters A8 and A22 redundant internal loads were determined by use of the Method of Dummy-— Unit Loads ~ a variant of the Least Work

theorem while the method 1S perfectly general, it becomes increasingly difficult to obtain Satisfactory accuracy as the degree of redundancy of the system increases To some extent, accuracy may be retained by skillful choice of redundants and through the use of carefully chosen determinate stress distributions (see Art A6.12, Chapter AS) A high degree of engineering Judgment must he exercised, how- aver Even so, it has been found difficult to apply successfully these Least Work methods to the low aspect ratio multi-spar wing

Several authors (2), (3), (4) have presented methods of analysis for the nighly redundant wing based upon use of the "Displace-~ ment Method" of analysis These references differ primarily in the techniques advocated

* Numbers in parentheses refer to bibliography at end of chapter

for the breakdown and idealization of the

structure into primary elements In the following example, a method following that proposed by Levy (2), is applied to an idealized delta wing structure The choice of this method for presentation is primarily for pedagogical reasons ~ it being the least detailed and consequently the easlest to grasp conceptually The reader who is interested in actual application is recommended to Reference (4) for techniques which are probably better able to handle practical problems because of their greater generality, flexibility and growth potential

A23.2 Basis of the "Method of Displacements” The Method of Displacements draws its name from the fact that displacements rather than forces are dealt with as the independent variables In earlier chapters the relation between structural deflections and applied loads was written through the use of the flexibility

influence coefficients as (see Eqs 24, 26,

Chapter A7)

(s)* [=]f-

where ơm 1s the đeflectlon of point m and Py is the external load at point n If one forms the

inverse of ann | » this equation may be re-

written as

(7) Bar} fi] 4} -

where [Ka | = [Ana] 1s called the stiffness

matrix

‘nile the flexiotlity influence coefficients

give the deflections per unit load, the stiff~

ness matrix gives the loads per unit deflection, Thus, any one column (say the mÈ5) of Em

gives the values of the forces (reactions)

developed at all numbered points of the structure when she corresponding point (point m) is de-

flected a unit amount, all other points held fixed

Trang 17

A23.2

quadrilateral "torque boxes" connecting points of intersection of the spars and ribs This

Fig A23.1 Idealized Delta Wing Structure

idealization {s discussed in more detail in the example Assume that the points of intersection have been mmbered as in Fig Ae3.2

Py Pha,

i

Fig A23.2 Reactions Developed at a Net of

Points when a Single Point is

Displaced

If one point, such as point 6, is dis-

placed a unit amount, the other points remain-

ing fixed, reactions are developed at these various points as shown These reactions may

include moments as well as forces, should rotations have been taken as pertinent displacements in addition to the translations (see

points 5 and 7 of Fig A23.2) The values of reoctions so developed would form the 6th

column of [k, land the 6th row, since (Knm] 18

always symmetric

Examination of Fig A23.2 shows that the resistance to the displacement of a single point is the cumulative resistance of 21] mem~ bers meeting at that point Thus, the stiffnesses are additive If the stiffness matrices

“* The resistance of a torque box having a corner at point 6 of

Fig A23 2 (say box 6-7-8-9) is a result of the tendency for the box to keep all four corners in the same plane Obvious- ly three-cornered ''boxas"' have no such resistance; hence

none were shown in Fig A23.1-

STRUCTURAL ANALYSIS OF A DELTA WING

are written for the individual elements of the structure, they are simply added to give the stiffness matrix of the composite Finally, the resulting stiffness matrix may be inverted to yield the matrix of flexibility infl

coefficients (Zq 2 "in reverse")

e

We note here, as an aside, that the Moment Distribution Method of Chapter All, the Slope Deflection Method of Chapter A12 and the Method of Successive Corrections of Chapters A6 and AlS are all examples of the "Displacement Method" of structural analysis In each of these methods the displacements are taken as the independent variables and these are adjusted to achieve equilibrium of the loaded structure The "adjustment" may appear as a systematic relaxation of artificial constraints (Moment Distribution and Successive Corrections) or it may be done mathematically in one stroke by the

solution of a set of simultaneous equations {Slope Deflection Method), The latter approach - solution of a set of simultaneous equations - Is essentially that followed herein, the "solution" being effected by matrix inversion

It 1s to be said of the Method of Displace- ments that it is complementary to the Least work method in that it 1s better suited to the

handling of the nighly redundant problem, while Least Work is better suited to problems of zew redundancies

A23.3 A Delta Wing Example Problem

The idealized structure of Fig A&3.1 will

be analyzed to determine

(a) the influence coefficients

(b) the internal stresses as a function of the applied loads

The grid points are numbered as in Fig 423.2 Note that the numbers increase to the rear and outboard

IDEALIZATION:

In the structure under consideration mem— ber 1-2-3-4 lies on the airplane centerline, The bending stiffness of this member includes that of the half-fuselage plus one-half of the ®ocarry-through structure." More detailed representations of the structure in this region are generally desirable, the oversimplified

model used here being employed to Limit the ‘ amount of data to be handled Some techniques which may be applied in idealizing the structure in this region are given in Reference (5)

Idealization, particularly with regard to effective skin areas, has been discussed in some detail by Lavy (2) The complexities of

Trang 18

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES this phase of the problem are too great to per=

mit an expansive treatment nere.* Sriefly, Levy's recommendations are:

(a) all skin may be considered effective

between spanwise spars when computing the cap

reas of such spars This assumption is sub-

ject to modification, of course, if spanwise

stresses are anticipated which will buckle the skin

(o) for streamwise ribs an effective width of 0.362L, where L is the rib length, may be taken as acting with each rib cap (Fig A23.3a)

(c) for the leading edge spar an effective width of skin of 181 of the span between spanwise spars is taken as acting with the cap of each such spar segment (Fig A23.3b)

Fig A23.3 Effective Widths of Cover Sheet for Rib and Oblique Spar Caps (After Levy, Ref 2)

The properties of the structure (Fig

A23.1) after tdealization, are summarized as foll#ys:

Net Point Beam Depth

{see Fig 425.2) (inches ) Le +- ee nee 5.86 2 -+ - 9.12 Be+ - 7.80 _.- 3.16 5 —„~ TT —~ 4.88 6 -+ - 7,86 Paeeee - ee 3.06 B - 4.02 9 -+ - 2.74 0-+ - 0.72 * A rational, systematic means of treating cover sheet panels is given in Reference (4) A23.3 Beam Element Properties

(Moments of inertia, in (inches*), are assumed to vary linearly between numbered points} 1/2 Fuselage Beam (1-2-3-4) Spar 2-5 - 3@48"- - —48"-— I= 2600 2700 3400 2200 64.91 18.25 Rib 5-6-7 Spar 3-6-8 ~ 48" 48" 49" 48" E———¬ CD 1=9.244 20.79 3.59 47.23 40.82 12.33 Rib 9-9 Spar 4-7-9-10 —48— —3648" — -— ea eS 1= 6.23 2.87 5.08 4.78 3.82 0.26 Leading Edge Spar (1-5~-8-10) 3961.8f2—m—— —— T= 16.02 12.29 8.30 0, 26

SIGN CONVENTION AND NOTATION:

The sign convention and notation adopted in conjunction with the grid numbering scheme of Fig A23.2 ts as follows:

Forces:

P, - transverse force at joint m, + UP

My - moment at joint m acting about 4 pitching axis, + NOSE UP

- moment at joint m acting about a rolling axis, + RIGHT WING UP

Trang 19

A23.4 Displac ements: 4n - transverse displacement of jotnt m, + UP @, - rotation of joint m about a pitching axis, + NOSE UP

G, ~ rotation of joint m about a rolling

axis, + RIGHT WING UP

Tm - rotation of joint m in the plane of an

oblique member, + IN DIRECTION OF + Qn (see page 423.3)

For a special purpose it will be convenient later to introduce another set of displacements to be called "relative displacements.” Where any confusion can exist, the displacements de- fined above, which may be visualized as being measured with respect to a set of reference axes

fixed in space, will be referred to as "absolute displacements."

With the above sign convention, any beam element (spar, rib or oblique member) which ts viewed so that its joint numbers increase from left to right, will have positive forces and displacements taken in the same sense as every other member This point is illustrated in Fig A23.4 Ms ta) ({f———¬ hibi-6-7 P, P, IP, aN (b} `: F———————_———— Spar 4-7-9-10 Py ?% 'Py “Pro q (c) roe rQs >Re Leading \ 4 Edge Spar ivy 'p, Py ÌP,,

Fig A23.4 Illustrating the Homogeniety

of the Sign Convention

CHOICE OF PERTINENT FORCES AND DISPLACEMENTS: Before beginning the analysis proper, the analyst must decide upon which forces and displacements are to be considered pertinent One will, of necessity, have to consider transverse forces (and translatory displacements) at all net points (ten points in the example) In

addition, rolling moments, N, (and rotations $)

must be considered along the airplane centerline

on all spars carrying across the centerline

Wherever three or more beam-like members inber~ sect (e.g points 5 and 8 of Fig, A23.2), their bending stiffnesses will react against each

other and hence couples in two planes (M, N) must be considered at these points.”

effects of various loadings are a, ponding forces or couples should be added the influence of additional forces at points intermediate to the grid intersections, may de accounted for by the addition of appropriate extra forces along beam spans Couples may be applied at any points where deflection slopes are required, ¢.2., the streamise slope at the trailing edge m t be needed in an aero-elastic analysis, in which case couples M,, M,, M, and Mi, (cf Fig Af3.2) would 2e employad The affect of couples from auxiliary aerodynamic surfaces and/or actuators may be desired, in which case appropriate additions may be made

In the present example, because of space limitations, only the minimum number of forces and corresponding displacements are considered,

viz:

P, through P,, - forces on net points

Ny Shrough N, - rolling couples on through- spars at the alrolage centerline

My, Ny» My, Ny, M, - pitching and rolling

couples at points of intersection of three beam elements A se

A23.4 Calculation of Element Stiffness Matrices

The task of computing the stiffness matrix for any one element of a configuration is a relatively straight-forward structural problem This problem may be either a statically determinate or a redundant one depending upon the

geometry of the element and upon the number of pertinent forces and displacements associated with it fit is a redundant problem, it is a

small one dy comparison with the overall structure of which it is an element One might say that a feature of the Displacement Method of Analysts is that it "makes (many) little ones

out of big ones"!

For instance, looking at Spar 2-5 of the present problem (Fig A2%3.Sa), we see that there are four forces (reactions) acting on it

* The implication here is that beam torsional stiffnesaes are not considered in such interactions This assumption is probably quite satisfactory in general, bending stiffnesses being much greater than torsional stifinesses for most beam elements On occasion, a beam will have consider- able torsional stiffness and it will then be necessary to account for it The leading edge spar might be such a beam if it replaces the "D" nose section of the wing in

the idealization

** Couples Q,, Q, and Q,, in the plane of the oblique Lead-

ing Edge spar, are inciuded hereby, since any such

couples may be resolved into M,, Ni; My, N,and M,, Ng

Trang 20

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES | | | { z _ụ a rd o 8 Fig A23.5

to be related to the four pertinent displacements at its ends That is, we seek the relation!

Na 6,

2 Š,

> * [k=] da

P 4, 4 6

between the loads (reactions) N and P and the displacements and A The first column of the above matrix relates the four loads to the

rotation $, at the left end with d,= 4,2 4,

= 0 The situation 1s shown in Fig A23.Sb where it is seen to be a familiar single redundant problem Thus, to obtain the complete stiffness matrix above requires the solution of four simple redundant problems, one for each column of the matrix

For Rib @9, on the other hand, we seek the relation

as may be seen from Fig A23.6, which depicts the calculation of che first column of this matrix, these are statically determinate calculations } @ t : en : iP iBy Fig A23.6

Obviously, one can proceed in the above fashion to work through all the elements of the structure solving for the stiffness matrix of each by application of these simple arguments Secause a large number of beam stirfness calculations of the above type appear in this problem, an alternate (but equivalent) procedure was developed for systematically treating these members This procedure ts detatled in the following section

Beam Element Influence Coefficients The method employed to obtain the beam

A23 5 element stiffness matrices involves first finding their flexibility influence coefficients and then inverting these matrices By xay of 111us- tration, the calculation of the influence co~ efficients is carried out in detail below for spar 4~7-3-10 (While the calculation 1s effected here by the Least Work matrix method, any of numerous other methods (Moment Area, Elastic Weights, etc.) may be employed at this stage, should the analyst see fit.)

Fig A@3.4b shows the forces applied to spar 4-7-9-10, corresponding to those considered pertinent for the wing (see above)

For the immediate calculation of influence coefficients, forces P, and P,, will be considered as fixed end reactions: deflections thus computed will be measured relative to the beam element ends To distinguish the deflections thus obtained they will be referred to as relative deflections and will be denoted by the

lower case Greek letters 6 and 9

Fig A23.7 shows the internal generalized forces for this spar

q q, 4z đ

=) 4 ==) 9 (E—' ` 9 qa, 1 0 q Fig A23.7 Internal Generalized Forces Used

in Influence Coefficient Calculation

for Spar 4-7-9-10

Member flexibility coeffictents for the tapered beam segments are computed next using the formulas of art A7.10 of Chapter A7 Collected in matrix form these become (Z£ = 1)

3.213 1.623 9 9

1.623 6.841 1.864 9

[a1] | 0 1.864 8.55 4.983

9 9 4.923 21.54

Next, unit values of the loads N,, P, and P, are applied successively to obtain the unit stress distribution: m t N, 1 Py Ø nh = -8 2333 9

Finally, the following matrix triple pro- is formed to give the influence coefficients Eq 24, Chapter A7)

duct

Trang 21

A23, 6 STRUCTURAL ANALYSIS OF A DELTA WING Cia) = a] Ba 10.20 -i2l.,6 ~-119,9 = 7141.6 2776 2386 -119.9 2566 3104 or, written out as an equation; For Spar 4-7-9-10 b, 10.20 -141.6 -119.3]/N, 5,)= | -141.6 2776 2566 |< P,)-(3) 6, -118.9 2566 3104 | [P,

Note again that for this beam

flexibility influence coefficients element the just computed are for relative deflections (note symbols used) The end transverse deflections

(A,, 4,,) and the end transverse reactions (P,, P,,) do not appear in these results

Beam Element Stiffness Matrices (Relative Deflections)

The next step involves finding the beam element stiffness matrices for relative deflections by inverting the above influence coerfi- elent matrix The results after inverting will be of the form

N 4

ey) = [x] (o} - (4)

N $

where i] may be called the "relative stiffness

matrix" Specifically, for Spar 4-7~9-10

N 3554 02308 =,005306 | | >,

PL pe 02505 -005015 -.001600 ;< 5, Py -.005306 -.001600 „001438 8,

" We note that the influence coefficient matrix to be inverted

does not appear to be ‘well conditioned” in the sense of Art AS 12 (see p A8 28} The situation is more apparent than real, however, and arises because of a difference in units among the coefficients of Eq (3) Thus, the appearance is

readily altered by "scaling" (factoring out appropriate con- stants) For instance, we may write oO, 19 20 714.16 711.96 Ny 6 the = | -14.16 27.76 25 66 10P, 6 afro 711.96 25 66 31.04 10P,

Scaling in this fashion does not actually enhance the con-

dition of the matrix (which is basically related to the size of the determinant of the matrix) but it does permit one to

assay the problem better

Complete Jeam Element Stifiness Metrices

The stiffness matrix given above r 2 the deflections relative to the beam element ends to the loads applied to the beam element, end reactions excluded It is desired next to obtain the "complete" beam element stiffness matrix in which absolute beam deflections are related to all the loads applied to the bean element including the end reactions

Consider deflecticns first It is desired to transform from the relative ceflections of Fig A23.8a to the aosolute deflecticns of Fig A23.8b S mer ner NG Ones | ———————— LBƯ3 (a) Om ®mvs me: = OT dm: > es (d) DATUM Fig 423.8 Geometry of {a} Relative and (b) Absolute Displacements From the geometry of has bn = br + =nts = on

the figure, one

Sper = - Hue Am + Am«p = SEE anes

dats = Snes + Sais on —

For comparison with the matrix form to follow, these equations are rewritten as

1

h xhh TT đ

Ser = SET Toy + Sper

bars = -+ Âm ot Anes

In matrix form these equations are written so as to provide a transformation from absolute to relative displacements In matrix form the index (subscript) is understood to increase monotonically down the column, and inasmuch as

Trang 22

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES A23.T

bn | "n3 (bn dy o| GL - -RIRT :-e x Na) wore snp [Nn Ẩm] wITHOUP END REACTIONS REACTIONS where 1 ° where L 9 2h TL I 1 1 6 HH mnrauưun + 1

Here ft] denotes a unit matrix.* The matrices are shown "partitioned", those portions which transform the end rotations being shown separ- ated from those which transform the transverse deflections

The transformation matrix (rj in 5q (5) 18 not square, having two more columns than rows to allow for the added two deflections, one at each beam end

Consider next the loads The applied loads may be transformed, using the equations of statics to yield the end reactions L Por M7 Sm+t_ 7 a ————————_) La a Mm+s Pm Pm+s Fig A23.9

From Fig A23.9 one has

Pq SR 2 Baer (Lo Igor) - SES

Fars = “B-+ aor Poer + up + yee

In matrix form, the general expression which intreduces the end reactions in terms of the other applied loads is

* We have used|_ _Jand

matrices, respectively (see Appendix A) Lọ - QJ to denote row and column Thus, is a row of zeroes

The above transformation matrix [s] is not square, having two more rows than columns, these extra rows yielding the end reactions

Provided the sign convention is observed as originally adopted, and provided the grid mumbering scheme is such that joint numbers increase to the rear and outboard, then the above transformation matrices will apply equally well to ribs, spars and oblique beams:

replace N by Mor Q, $ by @ or t**and ÿ by 8 or T

For the case of beam elements having couples applied along the beam (e.g the L.E spar,

Fig A25.4c), simple modifications of these transformation matrices are necessary

The relative stiffness equation, Eq (4),

is the relation between loads and relative deflections of a beam element Substituting from eqs (5) and (6), into (4), one obtains N H+ BILIE] Gp - 0 N WITH END ABSOLUTE REACTIONS DEFLECTIONS

Trang 23

A23.8

Eq (7) 1s the relation between absolute deflections and the complete loading on “he ele~ ment We let

Ee] = SJEJE]

be the "complete” element stiffness matrix

Note that (S] 1s the transpose of [T], as

it must be if [K] ts to be symmetric

To continue the 111uetrarion, the [7] ana

fs] matrices for Spar 4-7-9-10 are now written For this member the rotation at station 10 1s not considered and the couple at that end ts zero, hence the last row and column of [r] and [s] are omitted in writing them from the general expressions Equation (5) written for Spar 4-7-9~10: §, $,| |1 -.006946 0 9 006946| |A, 6,2 [0 -.667 1.0 0 ~.355 4, 6,| |0 -.5345 0 1.0 -.667 a, 4, Equation (6), written for Spar 4-7-9-10: IM, 1.0 9 bị P, | |-.006945 - 667 - 333 M, P, 9 1.0 9 P, Py 9 9 1.0 Py P „006945 - 33Z = 667 The matrix [x] 1s now formed per eq (8) For Spar 4-7-9-10: * Ns +3554 ~.01606 .0230 —.00140 | lỆ, | P ~.01606 0.3982 ~.001637 0,6252 0,165 ||a, | Py +02303 -.001237 005015 ~,001600 0,2258 ay Pe ~.005306 0,3262 -.001609 001438- ~.0, 4632] a, Pre -.001622 0,158 2282-0, 5638 san] cae

In the manner of the above illustrative case one goes through the calculations for the remaining beam elements to obtain:

* The notation Oy means there are N zeroes following the

decimal point and preceeding the first significant figure, e-g., 0,276 = 0000276, STRUCTURAL ANALYSIS OF A DELTA WING For Spar 2-5 Na 4.3c2 -.1218 1218 1.535 ', Py -,1218 2004208 -.004208 ~.O8054 A, P,| | 1218 -.002208 004208 08054 a, N, 1,555 -.0805¢4 08054 2,317 §, .01726 -~.0665+ My 158.5 -4,312 5.6580 ~1.535 ,268Z 8 PL W4.312 1557 9 -.2385 ,09995 -.01718 ||A, Pa) 560 ~.2385 4542 ~.2701 ,07429 là, Py -1.533 09993 ~.2701 2729 ~.1027 ||2, PL +2652 -.01718 07429 -.1027 94560 [i 4, For Rib 5-6-7 Me +0275 Pal ~.301294, P, ~.001204 ,001582 P 06886 For Rib 8-9 Ma +3322 ~.006910 09€910 8, Paps f-.006910 0,1437 -.0,1487 A, Py +006910 -.0,1437 0,1437 a, For Leading Sege Spar

** Because of its size, this matrix relation (and some to

follow) is written in a condensed tabular form, The correct

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ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES Transformation of Matrices for Oblique Beams:

The complete stiffness matrix of any beam element not either parallel to or normal to the streamwise direction, such as the leading edge spar in the present example, requires an additional modification to yield expressions relat- ing couples and rotations about pitching and rolling axes, rather than in the plane of the member Such a modification is readily made

In the present example, a pitching rotation

of joint 5 (@,) results in a rotation of joint 5

in the plane of the leading edge spar given by T, = sin A @, = 707 6,

where A is the sweep angle of the spar Like-

wise, a rolling rotation of jetnt 5 (9,) has a

rotational component in the plane of the leading

edge spar given by

Tl = cos A 0, = 707 G,

Then, when joint 5 experiences both pitching and rolling rotations, the total rotation in the plane of the leading edge spar is

T, = 707 8, + 707 6,

This last equation, and similar ones for joints land 8, when out in matrix form, yield the matrix transformation for the displacements A23.9 Next, consider resolution of the in-plane beam ¢lement couples Couples 2,, Q, and Q, have components in the pitching and rollin directions For example, at joint 5 ⁄

M, = sin A Q, = 707 4,

and

N, =cos A Q, = 707 Q,

Similar relations for the other couples along this member lead to the matrix transformation for this element's loads: u,) [77 0 0 0 0 0 06 n,| |.707 0 0 0 0 09 0 P, 9ø 1 0 0 0 0 o || Py 0 0 21 6 0 oo o iP P, 9o 0 O09 2 0 0 0 lJP Ma 0 0 oO Oo 707 9 ữ l}P N, © 0 0 0 70 0 9 lÍ Mẹ 9s o 0 0 ư 70 9 {I%s N, o 0 0 9 70 0 | (Tse P “ ° 0 0 0 0 9 1 The above two matrix transformations - one

38 8.9.9.8 9906 for displacements and one for loads ~ (and note poe pg 9 6 0 0llš that they are the transpose of each other) are

SR fF 9 9 9 9 0||% now applied to the complete stiffness matrix S 9 Lo Oo GC 9 FG 0 |ậ% equation for the leading edge spar as given in 90.0 70/9 9 9 9 ||â% tabular form in the last section The opera- 9 0 0 0 90,707,707 0 , tion involves premultiplying by one transforma- 9 0 0 0 6 38 90 1 $ tion matrix and postmultiplying by the other,

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A23.10

Torque Box Stiffness Matrices

In the case of the torque boxes, the stiffness matrices may be computed directly (they are determinate) The following approximate analysis is suggested as being satisfactory for most torque boxes A more detailed analysis may be in order for boxes with extreme geome- tries An alternate method is presented in Reference (2) and an appropriate discussion may be found in Reference (6) A completely different treatment of the cover skin is given in Reference (4) Consider the quadrilateral box of Pig A23.10 — ‘Rep: ‘esentati E E T: HH ive TH Section Fig A23.10

For purposes of this immediate analysis, the box 18 considered cantilevered from one end, such as r-s An elastic axis (e.a.) is

assumed to exist midway between the long sides

and an "effective root" is employed The torque on the box is

T = Py Om ~ Py Dy

An (assumed) average rate of twist, 9, is computed approximately by using the GJ at a repre- sentative section half way along the box."

9 =-2 - Baba GJ - Padn Gs

Then the deflection 5n and 6, are given by

ơn = OLnbn = Eno = Pata LyPn

ðm = GLubm = Prbm - Pnbn Luồn

By summing moments about the effective root, Py is found in terms of Py and then eliminated from the above equations to give

Lubm + imdn

= Palnbn

on = Gr ry

* Calculation of this GJ is discussed later The symbol "9"

used in these few equations bears no relation to that appear-

ing in the main body of the method

- PaLmÐm LnBm + Lmbn

on = Gs Ln

Since it is only the deflection of one corner (say, point m) which is to be related to the corner reactions, the box is now rotated about the effective root axis to reduce 6, to zero, The resulting total deflection of point m 18 = Ln Saporan = Om * Tr Sn E Ly * 0 = Palm apn ~ La t) ba + Dm) Re-solving this, we write z—B Ln

Pm “Tín + bm) (Tuồn + luồn) TTỢPAL

This last expression relates the load at point m to the deflection at that point, with the other three corners undeflected The reactions developed at these other corners are now found in terms of Py using the equations of statics (summation of forces and summation of moments about two axes) The result, expressed in matrix form is Fa | 2 Poi _} 7ha/Ln Py a, f ?a Ps Aa where A, = lnbm + Lmbn + bs (Lm - bn) * n(yp + Og A = xŒnồn + Lubn) + br (Lm - La) Ln (br + 5s)

IZ now the corner points have absolute displacements 4m, 4n, dr and dg, one can find from the geometry of the unit that the total relative displacement of point m may be written

Sonora, = [L 1 “min AA a |

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