Ni\Cf\L'3lc8 ARR No. 3J02 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED October 19^3 as Advance Restricted Report 3J02 STRESSES AROUND RECTANGULAR CUT-OUTS IN SKIN-STRINGER PANELS UNDER AXIAL LOADS - II By Paul Kuhn, John E. Duberg, and Simon H. Diskin Langley Memorial Aeronautical Laboratory Langley Field, Va. UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY P.O. BOX 117011 GAINESVILLE, FL 3261T-7011 USA a NACA WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. L - 368 yxi o<* ( " 3*©c«/* co vO I NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE RESTRICTED REPORT STRESSES AROUND RECTANGULAR CUT-OUTS IN SKIN-STRINGER PANELS UNDER AXIAL LOADS - II By Paul Kuhn, John S. Duberg, and Simon H. Dlakin SUMMARY Cut-outs in wings or fuselages produce stress con- centrations that present a serious problem to the stress analyst. As a partial solution of the general problem, this paper presents formulas for calculating the stress distribution around rectangular cut-outs in axially loaded panels. The formulas are derived by means of the substitute' stringer method of shear-lag analysis. In a previous paper published under the same title as the oreoent one, the analysis bad been based on a substitute structure containing only two stringers. The present solution is based on a substitute structure containing three stringers and is more complete as well as more accurate than the previous one. It was found that the results could be used to improve the accuracy of the previous solution without appreciably reducing the speed of calculation. Details are given of the three-stringer solution as well as of the modified two-stringer solution. In order to check the theory against experimental results, stringer stresses and shear stresses were meas- ured around a systematic series of cut-outs. In addi- tion, the stringer stresses measured in the previous in- vestigation were reanalyzed by the new formulas. The three-stringer method was found to give very good accuracy in predicting the stringer stresses. The shear stresses cannot be predicted with a comparable degree of accuracy; the discrepancies are believed to be caused by local deformations taking place around the most highly loaded rivets and relieving the maximum shear stresses. INTRODUCTION Cut-outs in wings or fuselages constitute one of the most troublesome problems confronting the aircraft designer, Because the stress concentrations caused by cut-outs are localized, a number of valuable partial solutions of the problem can be obtained by analyzing the behavior, under load, of simple skin-stringer panels. A method for . finding the stresses in axially loaded panels without cut- outs was given in reference 1, which also contained sug- gestions for estimating the stresses around cut-outs. In reference 2, these suggestions were put into more definite form as a set of formulas for analyzing an axially loaded panel with a cut-out (fig. 1) • CO 1-3 Skin-stringer panels, although simpler than complete shells, are highly indeterminate structures. In order to reduce the labor of analyzing such panels, simplifying ro> assumptions and special devices may be introduced.. The most important device of this nature used in references 1 and 2 is a reduction of the number of stringers, which is effected by combining a number of stringers into a sub- stitute single stringer. In reference 2, this reduction Y = i/Ki 2 + ?'- 2 - 1 - •''' 2 (Gfi 3 Young's modulus of elasticity, kips per square inch G shear modulus, kips per square inch Gt - Ju 2 V A 1 + ^2 8 ct l M ] \ = 3*1 \*1 + ~~) 8 K 2 - Eb 2 [A 2 A 3 ) CO I fi-t-.. K- = 3 "' SbiA<- Gt- K„ = 4 " Eb A 2" 2 K i o I" V K i P 9 N5 A 4 L half -length of cut-out, Inches (fig. 2) r 2/tr 2 > 2] P = k l \ 1K 1 " K 2 I ^l( X l 2 " ^2 2 ) P 2 = K 3 K 4 \l(\l 2 - \ 2 2 ) P^ = *i 2 (*i 2 - x x g) ^2^1 2 - >^2 2 ) K 5 K 4 *2^1 2 " ^2 2 ) Q-j_ force A-^o acting, on stringer 1 at rib, kips 0,2 force AoO acting on stringer 2 at rib, kips R stress-reduction factor to take care of change in length of cut-out (fig. 4) Xr difference between actual force in A^_ (or Ag) at the rib and the force Qj (or Q 2 ) , kips a width of net section, inches (fig. 6) b half -width of cut-out, inches (fig. 6) b-^ distance from Ag to centroid of An (fig. 2) b£ distance from Ag to centroid of A3 (fig« 2) T 2R t 8 r- r 2 r 3 A 3 a o T 2 E t 2 ~ T lR fc l A 2 (°2 R --0) '• R SboTo„ ^ ^R t]_ thickness of continuous panel, inches (fig. 2) tg thickness of discontinuous panel, inches (fig. 2) x spanwise distances, inches (For origins, see figs . 2 and G. ) y chordwise distances, inches (For origins, see fig. 2.) H"\ \ v 2 + K 2 2 + \/fe 2 + Kg £ ) 2 - iX 2 i/ 2 / Kl 2 + % 2 - \^i e + K 2 K ) £ - 4^7'° o Q average stress in the gross section, kips per square inch CQ a-j_ stress in continuous substitute stringer, kips per N ^ square inch Og stress in nain continuous stringer, kips per square inch 0-5 stress in discontinuous substitute stringer, kips per square inch a rib stress in rib, kips per square inch o average stress in net section, kips per square inch t 2_ shear stress in continuous substitute panel, kips per square inch Tg shear stress in discontinuous substitute panel, kips per square inch Superscripts on stresses denote forces producing the stresses. Subscript R denotes stress occurring at rib station. Tensile stresses in stringers are positive. If the center line of cut-out is fixed, positive shear stresses are produced by a tensile force acting on A^. Simplified Three-Stringer Method A principle for the effective use of substitute stringers was stated in reference 3 substantially as fol- lows : 10 Leave the structure intact in the region of the stringer about which the most important actions take place, and replace the stringers away from this region by substitute stringers. In a panel with a cut-out, the most important action takes place around the main stringer bounding the cut-out. In accordance with the foregoing principle, the three-stringer method is based on retaining the main stringer as an individual stringer in the substitute structure; one substitute stringer replaces all the remaining continuous stringers, and another substitute stringer re- places all the discontinuous stringers. The three-stringer substitute structure obtained by this procedure is shown in figure 2, which summarizes graphically the salient features of the method. The figure shows the actual structure, the substitute structure, and the distribution of- the stresses in the actual structure. The maximum stringer stress as well as the maximum shear stress occurs at the rib station. The formulas given hereinafter for the stresses at the rib station and in "che net section are based on the exact solution of the differ- ential equations presented in appendix A. The formulas derived from this exact solution for the stresses in the gross section are somewhat cumbersome and are therefore replaced here by formulas that are based on mathematical CO I hi 11 approximations of sufficient accuracy for design work (appendix B) . The use of these approximations is the reason for calling this method the simplified three- stringer method. Stresses at the rib station in the substitute structure . - The stringer stresses at the rib station are °i R = °(i -TT (1) c 2r = a(l + RC Q ) (2) where the factor C , for a cut-out of zero length, is obtained from figure 3 and the factor R, which corrects C for length of cut-out, is obtained from figure 4. For practical purposes, the parameter 3 appearing in figure 4 may be assumed to equal unity. (See appendix A.) The length factor R depends, therefore, chiefly on the parameter K-jL. This parameter is roughly equal to L/a for usual design proportions; in other words, the length effect can be related more directly to the length-width ratio L/a of the net section than to the proportions of the cut-out itself. The running shear in the continuous panel at the rib station is 12 ]_ t-i = - ORCoAg^i tanli K^L (3) The running shear in the discontinuous panel at the rib station is K 4 / % 2 \ T2 R t 2 = " *A 2 - (l + RC + — ) (4) in Which the factor D nay be obtained from figure 5. The stresses ov and t „ are the maximum values of <*R ^R Oo and Tg, respectively, and are the maximum stresses in the panel. The stress a-, reaches its maximum at the center* line :>f the cut-out. The stress t-j_ reaches its maximum in the gross section at the station where <*1 = ^2 1 c o' Stresses in the net- section of the substitute structure . The formulas for the stresses in the net section are ob- tained from the exact solution (appendix A) . At a dis- tance x from the center line of the cut-out, the stresses in the continuous stringers are / RC^Ao cosh Kix\ o\ = o T, = O I 1 _ - 1 l x A 1 cosh h-,1 _ / cosh K-iX\ As the length of the cut-out - or, more precisely, the 13 length of the net section - increases, the magnitude of the parameter K-,L increases and the stresses a-. and o r? converge toward the average stress a in the net section. The running shear in the net section is _ sinh K-,x T n t, = - CRC A Q K, — 1 1 o 2 1 C osh K-jL and decreases rapidly to zero at the center line of the cut-out. Stresses in the gross sec tion of the substitute structure .- The stresses in the gross section can be obtained from the exact solution given In appendix A, but the for- mulas are too cumbersome for practical use. A simple ap- proximate solution can, however, be derived (appendix 3) that gives good accuracy in the immediate vicinity of the cut-out and reasonable accuracy at larger distances from the cut-out. The approximate solution assumes the differ- ences between the stresses at the rib station and the average stresses in the gross section to decay exponentially with rate-of-decay factors adjusted to give the initial rates of decay of the exact solution. The stress in the cut stringer by the approximate solution is c 3 = o Cl- e" r l X ) (5) 14 The stress in the main stringer is 2 = o + (0 2r - o )e 2 (6) The stress in the continuous stringer 1 follows from statics and is °1 = °o + if C°0 - °2) + if (°o - °c) (7) The running shears in the sheet panels are -rnX , , -rox T X t! = T £R t 2 e - - (j £R t 2 - T lR t^e (8) T 2 t 2 ~ T 2 R t 2 e ^ - 9 ' Stresses in the actual structure .- By the basic prin- ciples of the substitute structure, the stresses in the main continuous stringer of the actual structure are identical with the stresses in stringer 2 of the substitute structure; the total force in the remaining continuous stringers of the actual structure is equal to the force in stringer 1 of the substitute structure, and the total force in the cut stringers of the actual structure is equal to the force in stringer 3. In the shear-lag theory for bear.s without cut-outs (reference 1), the force acting on a substitute stringer is distributed over the corresponding actual stringers on 15 the assumption that the ehordwise distribution follows a hyperbolic cosine law. Inspection of the test data for panels with cut-outs indicated that neither this nor any CO k> other simple assumption fitted the data on the average as well as the assumption of uniform distribution. It is therefore recommended, for the present, that the stresses in all continuous stringers other than the main stringer be assumed to equal a-j_ and that the stresses in all cut stringers be assumed to equal o 7 . (See fig. 2.) The validity of these assumptions will be discussed in con- nection with the study of the experimental data. Again, by the principles of the substitute structure, the shear stresses t -, in the substitute structure equal the shear stresses in the first continuous sheet panel adjacent, to the main stringer. In order to be consistent with the assumption that the chordwise distribution of the stringer stresses is uniform, the chordwise distribution of the shear stresses should be assumed to taper linearly from T i to zero at the edge of the panel (fig. 2). Similarly, the chordwise distribution of the shear stresses in the cut sheet panels should be assumed to vary linearly from T2 adjacent to the main stringer to zero at the center line of the panel. Inspection of the test data indicated that this assumption does not hold very 16 well in the immediate vicinity of the cut-outs. The dis- crepancy is of sone practical importance because the maxi- mum stress in the rib depends on the chordwise distribution of the shear stress at the rib. By plotting experimental values, it was found that the lav/ of chordwise distribution of the shear stress t.~, at the rib station could be appro;-:!- mated quite well by a cubic parabola. The effect of thin local variation may bo a.Toumed to end at a spanwise distance from the rib equal to one-fourth the full width of the cut- out. A straight line is sufficiently accurate to repre- sent the spanwise variation within this distance (fig. 2). If the stress t,-, is distributed according to cubic law, the stress in the rib caused by the shear in the sheet is T _t b H -n (j rib 4i-. rib i - if f (10) •fled Two-Stringer Method e two-stringer method of analysis given in reference 2 is more rapid than, but not so accurate an, the three - stringer method previously described. It was found, how- ever, that some improvements could be made, partly by in- corporating, some features of the three-stringer method, partly by other modifications. 17- The nain features of the modified two-stringer method are summarized in figure 6. The cut stringers are re- placed by a single substitute stringer; and all the uncut ^ stringers, including the main one, are also replaced by a single stringer. Contrary to the usual shear-lag method, however, the stringer substituted for the continuous stringers is located not at the centroid of these stringers but along the edge of the cut-out. The substitute structure is used to establish the shear-lag parameter K, which determines the maximum shear stress, the spanwise rate of decay of the shear stress, and the spanwise rate of change of stringer stress. The maximum stringer stress nust be obtained by an independent assumption, because a single stringer that is substituted for all continuous stringers obviously cannot give any indication of the chordwise distribution of stress in these stringers. No solutions are obtained by the two-stringer method for the shear stresses in the continuous panels, either in the net section .or in the gross section. Stresses in the substitute structure .- Throughout the length of the net section, the stress in the main stringer is q 2r = O P- + 2R(C - 1)1 (11) where C is the stress-concentration factor derived in 18 reference 2. Values of C may be obtained from figure 7, which is reproduced from reference 2 for convenience. It may be remarked here that reference 2 placed no explicit restriction on the use of the factor C; whereas the use in formula (11) of the correction factor 2R, which varies from 2 for short cut-outs to 1 for long cut-outs, implies that the factor C by itself should be used only when the net section is long. In the gross section, the stress in the main stringer decreases with increasing distance from the rib according to the formula °2 = °o + (°2 R " °oK KX (12) The stress in the discontinuous substitute stringer is 03 = o (l - e - ]Lx ) (13) The stress Ch nay be obtained by formula (7) when o and 03 are known. The running shear in the discontinuous panel is given fey T 2 t 2 = - a A 3 Ke- IO: (14) Stresses in the actual structure .- The stresses in the actual structure are obtained from the stresses in the sub- stitute structure under the same assumptions as in the three-stringer nethod. 19 EXPERIMENTAL VERIFICATION OP FORMULAS AND COMPARISON OF METHODS c0 Test Specimens and Test Procedure i In order to obtain experimental verification for the formulas developed, a large skin- stringer panel was built and tested. The panel was similar to the one described in reference 2, but the scope of the tests was extended In two respects: Very short cut-outs were tested in addition to cut-outs of average length, and shear stresses as well as stringer stresses were measured around all cut-outs. The general test setup Is shown In figure 8. A setup of strain gages Is shown in figure 9. The panel was made of 24S-T aluminum alloy and was 144 inches long. The cross section is shown in figure 10(a) ; figure 10(b) shows for reference purposes the cross section of the panel tested previously (reference 2) . Strains were measured by Tuckerman strain gages with a ga^e length of 2 inches. The gages were used in pairs on both sides of the test panel. Strains were measured at corresponding points in all four quadrants. The final figures are drawn as for one quadrant; each plotted point represents, therefore, the average of four stations or eight gages. The load was applied in three equal increments. If the straight line through the three points on the load-stress 20 plot did not pass through the origin, the line was shifted to pass through the origin; however, if the necessary shift was more than 0.2 kip per square inch, a new set of read- ings was taken. , 3 An effective value of Young's modulus of 10. lo * 10- i per square inch was derived by measuring the strains in all stringers at three stations along the span before the first cut-out was r.ade . This effective modulus nay be con- sidered as including corrections for the effects of rivet holes, average gage calibration factor, and dynamometer calibration factor. The individual gage factors were known to be within ±~ percent of the average. The average strain at any one of the three stations in the panel without cut-out did not differ by more than 0.05 percent from the final total average. The maximum deviation of an individual stringer strain from the average was 5 percent; about 10 percent of the points deviated from the average by more than 2 percent. A survey was also made of longitudinal and transverse sheet strains at one station near the center. The average longitudinal sheet strain differed from the average « stringer strain by 0.05 percent. The average transverse strain indicated a'Poisson' s ratio of 0.523. 21 Discussion The results of the tests are shown in figures 11 to 33. Calculated curves are given both for the exact three- stringer method and for the simplified three-stringer method. It may be recalled that either method assumes that the stresses in all continuous stringers except the main stringer have the magnitude c-j. anra i n a ^- cu ^ stringers, the magni- tude CTg. Because the values of On and c^ do not dif- fer very much for the two methods, the curves for them com- puted by the simplified method are drawn only once in each figure. A qualitative study of figures 11 to 32 indicates that the stress distribution calculated by the theory agrees sufficiently well with the experimental distribution to be acceptable for most stress-analysis purposes - in particular, the maximum stresses in each panel agree fairly well with the calculated ones, The most consistent discrepancies are chargeable to the simplifying assumption that the stringer stresses are identical in all the stringers repre- sented by one substitute stringer. As a result of this assumption, the calculated stresses tend to be too low for the stringers close to the main stringer and too high for the stringers near the center line and near the edge of the panel. The fact that the calculated stresses for some of 22 the cut stringers are lower than the actual stresses is of little practical importance because these stringers would probably be designed to carry the stress a rather than the actual stresses. On the uncut stringers, however, it may be necessary to allow some extra margin in the stringers near the main one. Aside from the consistent discrepancies just noted, figures 11 to 52 show that the stresses in the main stringers sometimes decrease spanwise more rapidly than the theory indicates. It is believed that this discrepancy also will seldom be of any consequence in practical analysis. Of paramount interest to the analyst are the maximum values of the stresses. The quantitative study of errors in the maximum stresses is facilitated by table 1. The highest stresses occur theoretically at the rib station but, for practical reasons, measurements had to be made at some small distance from this line. The comparisons are made for the actual gage locations. The calculated values for the three-stringer method are based on the exact solu- tion but, in the region of these gage locations, the exact solution and the simplified solution agree within a fraction of 1 percent. The errors shown by table 1 for the maximum stringer stresses computed by the three-stringer method are but 23 little larger than the local stress variations that were found experimentally to exist in the panel without cut- out. Presumably these variations are caused largely by failure of the rivets to enforce integral action of the structure. The errors in the maximum shear stresses computed by the three-stringer method are consistently positive. The discrepancies ■ are possibly caused by the sheet around the most highly loaded rivet3 deforming and thereby relieving the maximum shear stresses. The errors are higher than those on the stringer stresses and corrections to the theory appear desirable in some cases. The criterion that determines the accuracy of the theory cannot be definitely established from the tests. A rough rule appears to be that the error increases as the ratio of width of cut-out to width of panel decreases. The errors given in table 1 for the two-stringer method show that this method is decidedly less accurate than the three-stringer method for computing maximum stringer stresses but that there is little difference between the two methods as far as the computation of the maximum shear stresses is concerned. A general study of the two theories indicates that this conclusion drawn from the tests is probably generally valid. It may be 24 recalled here that the two-stringer method gives no solu- tion for shear stresses in the continuous panels. Comparisons of the maximum observed rib stresses and the computed stresses are given in table 2. Two values of computed stress are shown. The smaller value was obtained on the assumption that the filler strips between the ribs and the sheet were effective in resisting the load applied to the ribs; whereas the larger value was obtained on the assumption that the filler strips were entirely ineffective. In figure 23, the chordwise variation of the observed and computed rib stresses is shorn for three cut-outs. Because the chordv/ise distribution of shear stress in each sheet panel between two stringers is essentially constant, rib stresses computed by formxila (10) will be too snail v/hen only a few stringers are cut. The computed values of rib stress were therefore determined by calculating the shear stress at the center of each panel according to the cubic lav/ and assuming this shear stress to act in the whole panel. The agreement between calculated and observed rib stresses is not all that could be desired. The discrepancy may be attributed to the approximation used for determining 25 the shear stresses and the uncertainty of the effective area of the rib. •^ Langley Memorial Aeronautical Laboratory, >J Nation. 1 ! Advisory Committee for Aeronautics, Lai.gJey Field, Va. 26 APP3' T DIX A KXACT SOLUTION OF THREE- STRINGER STRUCTURES For a two-stringer panel constituting one half of a symm*trical structure, the application of the basic shear- lag theory yields the differential equation p r\-'J O ~ - K°T = (A-l) which Is given in slightly different form in reference 4. In the analysis of a skin-Stringer panel with a cut-out, a three- stringer substitute structure is used. (3^e fig. 2.) Application of the basic equations of reference 4 to a three- stringer structure yields in place of the single equa- tion (A-l) the simultaneous equations d tJ T dr. I " K lS + K 3 T 2 = ° fj2 dx 2 - Kg Tg + K4T X - (A-2) On the simplifying assumption that the panel is very long and that it is uriifo-iT.ily loaded by a stress O at the fur ends, the general solution of the equations (A-2) is -'hX -A, x T l = c l e + c 2 e (A-3) To ±) -\-,X in wh±ch c-i and c 2 are arbitrary constants. 27 Because the structure is assumed to be symmetrical about the longitudinal as well as the transverse axis, CQ the analysis may be confined to one quadrant as shown in V figure 34(a). The analysis can be simplified somewhat by severing the structure at the rib and considering sepa.rately the net section and the gross section. The solutions for the two-stringer structure representing the net section can be obtained from reference 4. The solutions for the three-stringer structure representing the gross section are obtained conveniently by considering two separate cases of loading. In the first cane, stresses a are assumed to be applied at the: far end, and the stresses at the rib station are assumed to be uniform and equal to the average stress a necessary to balance the stresses a . The forces at the rib station existing in the stringers are called the Q-forces (fig. 31(b)). In the second loading case, a group of two equal and opposite forces is assumed to load the stringers 1 and 2 at the rib station. These forces are called X-forces (fig. 34(c)). In the net section the boundary conditions are as follows: At x = 0, t-j_ = (from symmetry) At x = L, XR °1 " " A X 2 " Ag The substitution cf these conditions in the solution of eo.ua tion ( A- 1 ) yi e 1 ds v XoKi s?'nh K n x v T, - - — - : — - [J\-c ) \ 1 t] cosh K-jL ." □ >sh K-\X a a ,, _ __- ^. (A _.- ■i .-, cosh K.-.L ' cosh K-iX a - = S -L. (a-7 The superscript X ind that the stress.?,?, are those caused by the action of the X-foroes. In order to obtain the total stresses, the average stress o* must be added to J, X or Or ^. rn he shear stress T^X is the total stress because the uniform str a is not accompanied by any shear stre sn the ^-forces ore .ed to the cross section, the be i i, ions at x = are "^2 — Cl - v~ Oo - — - O Ot-0 1 A l - A 2 -5 29 Applying these conditions to equations (A-3) and (A-4) gives the following solutions for stresses: Q* -\,x -'MA -A. X T 1 Q = " ( P 2 e ' " P 4 e & V> = tt 7 (A-8) — — — — — — — p -l _ — — : p. & K, *3 2 o Ag 1 ° " ^ V^ e ' x 2 e / .^2 [ P 2~ *2 2\ t x K 3 rVl a 3 > anc ^ To_, obtained by the exact solution, and the corresponding average stresses in the gross section decay exponentially with rate-of --decay factors adjusted to give initial rate3 of decay equal to those of the exact solution. These rates can be written simply in terms of the stresses at the rib and the properties of the panel. The solutions for Jq_ and t-j are then derived from the solutions for On and a-*. If it is assumed that the stresses in the cut stringer can be expressed by 3 r-. aji - e~ Ti:: ) then da 3 -r n x dx~ = a o r l e ~ b t it , f rora the basic she ar- lag t he □ r y , da ? T 2*2 - r ,r. dF = - ~~T = a o r i e x (B - 15 Therefore, at x = 0. TV, tg ^R _ 33 The stress in the main continuous' stringer can be approximated by °2 = °o + (°2 R - o )e >r 2 : co N ^ which yields da -r x dF= -i% " %)^ 2 (B-2) but, from the shear-lag theory, da 2 T l t l T 2 fc 2 . % -r P x "sr = - ~xr + ~~ = - (a 2 " a o^ re * Therefore, at x = 0, /To to - t-i t n ) ^R ^ - L R 1 J r2 " * A 2 (o 2r - a Q ) The value of a^ can be obtained by statics from c<2 and Or and is o A A a l = a o + if (°o " *£.) + SJ <°o - °3.) ( B "3) If the value of To is assumed to decay exponentially, then " r 3 x and t q - To e 2 ^ R dT - -To r-?e ^ dx ^R ° but, from the shear-lag theory, dT 2 G " r 3 x dF = Ebi l°2 " a 3> = " T 2 R r 3 e 34 Therefore, at x = 0, Go r* =■ - = 2 R The shear stresses in the continuous panel can be determined fron the rate of change of o-^. Pror.i the shear- la r theory, 6.0-1 r 1 t 1 dx /i]_ Differentiation of formula (B-3) yields da. A, dc, Ar, &o n ~ = - 7^ T- - ~ —^ (3-5) dx A]_ dx A ^ ax Substitution of the derivatives (B-l) and (B-2) already obtained in (B-5) gives ^ = ^ (^o p - ajp.-V . £ o ori e- r l X (B-6) dx Ai \ ^R 0/ A]_ ° J- Pinally, substitution of (B-6) in (E-4) yields T l t l = T 2 R t 2 e - ( J2 R t2 ' TIr 1 "'!^ 35 APPENDIX C NUMERICAL EXAMPLE Analysis by the Exact Three-Stringer Method The structure chosen for the numerical example is the 16-stringer panel tested as part of this investigation. The particular case chosen is the panel with eight stringers cut and with a length of cut-out equal to 30 inches. This cut-out is the one shown in figure 8. The cross section of the panel is shown in figure 10(a). The basic data are: t 1 ' sq 7 n 0.703 ^2» sq in 0>212 '*> ?q m x.045 I 1 ' } n < 0.0331 l 2 ' } n 0.0331 ?!' } n 5.96 ^'. xn 7.56 L > in 15.0 These data yield the following values: Kf= 0.01295 K 2 2 = 0.00944 r.,3 = 0.00995 K 4 = 0.00785 K - 0.00664 Prom these parameters follow the factors for the rate of decay, which are ,/ k^ + Kg 2 + \J(K^ + K 2 2j 2 -I P A-l = y ^ = 0.1421 \ 2 = y SL^-i £L^ - 0.0467 36 The computations of stress nay more easily be made in terms of the constants P^, P 2 , p 3> and p 4> tne values of which are - K] - C - 1 '' " ^2*7 - -01205(0.01295 - 0.00218) _ Q QC ., 5 1 '" x a 2 , 2\ 0.1421(0.02021 - 0.00218) 1\ 1 " 2 ) K 3 K 4 0.00995 x 0.007 _ „,-_ p 9 = ; ~ = = 0.000b \t(\t 2 -\o^ 0.1421(0.02021 - 0.00210) J- \ -L O / _ K iH :: i" ~ x i 2 ) _ 0.012^5(0.01:.,.^ - o. •■. 021) _ _ 1117 V (\-, 2 - \n 2 ) 0.0467(0.02021 - 0.00218) P - '^"^ _ 0.00995 x O.00TC5 _ Q Q90? " 4 ^(^l 2 ~ W' 3 ) 0.0467(0.02021 - 0.00216) The reduced stress-excess factor is P, - Po RC„ = P x + P 2 - P 3 - ? 4 + K-, tanh K X L 0.0927 - 0.0505 n 2 o 6 0.0545 + 0.0305 + 0.1117 - 0,0927 + 0.1065 * Ilth a force of 7.5 kips acting on the half panel, °z = TT9T§ = 3 ' 82 ki P s /sq in. 37 and 77 = 0.9 J 7.50 „ , ° = n~"7~ = 8 » 21 kips/sq in. Therefore , X R = p .C ? ~Ao = 0,295 x 8,21 x 0.212 = 0.514 kip Stress e;j in the_n?J:_ aecti^qn . - The shear streps In the substitute panel of tv>, nee section is found by equation (A-5) •T X-pTn sinh KtX t-j_ ciosS K X L - Q-S-" 1 ^ x 0.1158 sinh 0.1158x 0.0551 cosh 1.707 - - 0.620 sinh 0.1138* At the rib station, x ~ 15.0 and t 1r = -0.620 sinh 1.707 = -1.65 kips/sq in. The stringer stresses are found by substituting in equations (A-6) and (A~7) and adding the average stress a - a - R —- ^- Pi 0'514 cosh 0.1158x 1 A{ cosh K]! ' " 0.703 cosh 1.707 = 8.21 - 0.257 cosh 0.1138x _ - ;% cosh i: x x 0.514 cosh 0.1158x '2 ' a + Ao cosh K-.L 0.212 cosh 1.707 = 8.21 + 0.850 cosh 0.1138X 38 The maximum stringer stress occurs in the main stringer at the rib, x = 15. The nearest gage location was at x - 15.5, where Co = 8.21 + 0.050 cosh 1.536 = 8. 21 + 2.05 = 10.26 kips/sqin. Stresses in the rross section .- The stresses in the gross section are obtained by adding the solutions for the stresses due to the X- and Q-forces, The shear stress in the continuous panel is obtained by adding equations (A-8) and (A-ll) . The final solution thus obtained is At the rib station, x = and TV = -.'.92 - 4.57 = -1.65 kips/sq in. R This value of T-, checks the one previously obtained for this sane station in the net section. Substituting the constants in equations (A-9) and (A-10) and combining gives T _ „ ., -0.1421x . __ -0.0467* t = -2.13e - 4.95e At x - 1.50, the point of maximum observed shear stress, To = (-2.13) (0.809) - (4.95) (0.931) = -6.33 kips/sq in. ■ 39 The stress in. the continuous- substitute stringer Is found by combining equations (A-10) and (A-12) . The final result is o 1 = 3.32- 0.97e-°- 1421X + 4 . G3e"° - 0467x Similarly, the stresses in the main stringer and in the cut stringers are found by adding, the proper values of the X- and Q-strespes. In the main stringer, _ _ RO -0.1421* . -0.0467:: and, in the cut stringers, o, = 3.82 - 0.4Ge-°- 1421x - 3. 3 6 e -°- 04S7j: Pints of the computed stresses in the panel for this cut-out are shown in figures 22 and 30. Analysis by the Approximate Three-Stringer Method The basic data are the same as for the exact three- stringer method. Compute K l g ' i: 2^ _ 0.01295 x 0. 00944 _ - K 5 K 4 ' 0.0099.5 x 0.00705 " * K 0.00664 ' = 0.704 K£ 2 0.00944 Prom figure 3, C G = 0.G00 40 From figure 4 for K-,L ~ 1.707 and the exact value of B = 1.10, there is obtained R = 0.492. The stresses in the continuous stringers at the rib are, by formulas (1) and (2), o, = 8.21 iR - (0.492) (0.600) (q' ^qI ) =7.48 kips/sq In, o - 8.21|1 + (0.492) (0.600) | = 10.63 kips/sq in. The running shear in the continuous panel at the rib is, by formula (3), t-i t-i = -0.21 x 0.492 x 0.600 x 0.212 x 0.1128 tanh 1.707 1 R l = -0.0547 kip/in. The maximum running shear in the cut panel is computed by formula (4). The value of D is obtained from figure 5j with K = 0.00664 and K-j 2 + K 2 2 = 0.02239, D = 0.109 and T 2R t 2 = -8.21 x 0.212 x %20||5 - -0.234 kip/in. 1 + (0.492K0.600H <$*§§§] T 2R = " Wri = - 7 -° 8 kil>s/sQ in - The stresses in the net section are computed as for the exact solution. 41 The rate-of-decay factors for the stresses in the gross section can now be computed T Q t<"> Pl A 2 o ~ 3.82 x 1.045 u ' U£ > b/ To to _ T-, tn ^ - R R - -0.254 h- 0.055 _ .„,,, 2 " Ao(0 2l3 - o.) " " 0.212(10.63 - 3.82) U -^~ D 1 ^R _ 0.5 80 x 10.65 _ n w __ 3 Ito2 T 2T? V,b6 -7.08 The stress in the cut stringers by formula (5) is 3 - 3.82 (l - e -0.0587x) and in the main stringer by formula (G) is , QO _ _, -0.1236X Og - 3.82 + 6..93e The stress in the continuous stringer can be found by formula (7) . The running shears are, by formulas (8) and (9), n „„„ -0.0587X _ ___ -0.1218x T 1 t 1 = -0.234e + 0.179e „ n , A -0.0755X T_t_ = -0.234e At x = 1.50, the point of maximum observed shear stress, T 2 to = -0.234 x 0.893 - -0.209 kip/in. and 42 0.209 T 2 0.0331 = -6.51 kips/sq in. alysis by the Two-Stringer Solution The basic data remain as before. Compute a _ 0.: b ~ 14.06 = 0.6 _ 1.045 v ?>30 b(A]_ + A 14.06 x 0.915 = 0.764 Then from figure 7 is obtained . C = 1.195 The maximum stringer stress can then be computed by formula (11) a 2 .= O.SlQl + 2(0.492) (0.195)] = 9.85 kips/sq in. By statics, a 3 ID £ .£5 ■a c i-i n O 10 r-l a CO «H O it u c O « ' »-H Qj a) -h BE "H B m B « cr u *h 9 n « K C\ ff) cti -u n jd E en a, O «-l t> M C c o •H «H ■u 3 0J DO c c «H O U -H ■*J *J m 3 I iH 41 U C o e~ c- o « W « a. T3 c « cr rH 03 r-l a. CJ .* >% c o c~ (=1 e c "O c ih a. as -h o a: a H n tf p n x> E oi a O «H •a o o. i xi a — htfO > rH 60 I G «J C *> tH K«3- £| .H o Eh *D'«» , ^wcor-ir#-i O>CM0JrHr-a>t--*J' CM rH I lOrHt>[->C-lOO cor-v^^^c-co o * m io i i i i i i i i r-rHtoc^to-^cftcji HOOHlOMOlO IO CM rH i-HrOOCDOtOCMC- cnOOllO^OiDt- lOlOlOtDlOlDMJ) MH«iOOOOH OlrHtOinOaiCM-* lDOIO»HOO>Oi IO CM rH t> Q> lO IO I rH I rH rH I I I OMOOOlOOO MOinoiococOf MS 01 O O U N (Jl CJlOrH^O^'-HOJTj' CJHWIDIOHHH CMCOC>tOIOlO'«">el" OOO'J'IOlOCMrHlo t-tOONOOrtOl CJO>rHOOl>OrH COtOt-CMOtOtOlO COCOOlOHOlOOJ oooooooo lOlOlOlOiOiOlOlO lOlOlOlOOOOO HHrtHCOlOlOlO WTflDCOCOCOOW •* co cm • ••••• en o> rH in co co CM III OOlMOlOrl « r-i t> r~ cm m ■* in in io co 10 cuvOHnm tO CM * C- rH t- CD rH 0> IO CM CO £> t- t- IO O) io <• m to co m , to * CM CO CM tO I m t> cc o o co to *»■ m o as in ■* ■*•*•»■* in Cfc O Oi CT> 0*s CD CO CO CO CO CD o IO to IO IO io to co co co cr* co co rH to in c- en o> m in in in in in IE 60 c *H t, ■U Or. I CC ■V c a! I rH CO i> m rHrH O O O O *t in co cm o> IO O) o cm to t- to rH CM o o o o c- oo t> to o o o o CM 'J" CO to to rH in IO i-< i ■ o o o o •w o m to t-i<3> rH 0> CM CM CM CM CM in m co i i i i OlOrtH to o> cm a> co c- o> c- ■H CM rH CM m o m o co * cm to co o> o o rH CJ CM tO o o o o CM CM CM CM o oo o m in m m •3 to as u CO > 45 to CO > 4> OJ ts » a) A a> C u •o c 1 a> c > -H ■c t. a •H ■d / 1 /////I o rH o II / 1 In //// II > II M o o £x^_ //// 4-* / iii rH °N/ / //// 1 *-. // i *£>/ / /Q 5- / M / / y ~y^- 5^2 / / {/ o ^f- oo CO vO CO •tf 00 CO co O oo Q0 o c_> vO CvJ 1_ CVJ c2 CVJ c\j CVJ CM is: Q. O (_ O Q 1 oo vO CvJ en CO vO OJ O o o NACA Fig. 4 on ^_^ c\j ID H cd o Cfl rO M — : •sr X. a rsj / / ■H / / / vO T3 ' / / O / / <\i / / II // M / / o / / o ' / CVJ rH Q / rH op • ^ ^ ^ <*\^ ^^ ^ >^ ^-^ ^^ O QC o e Q. E o i 0) (_ .£? li- en oo vO lO cc NACA Fig. 5 u o Q. O L ! lO *W Figure 6r5tress distribution around cut-out by modified two-stringer method. NACA Fig. ? O IHWJ i i i i I i i i i I 1 1 i oo C\J 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 op Q 00 sO £ o i_ u ? o NaCA Fig. 8 IT) rH II O fc>0 c 4-> W -P & •r-l 0) x: +> C ctf ft u Qfl ^4 +-> w I CD O I 0> W -u CD H QlJ NAGA Fig. 9 » Eh •H NACA .0987-n r t-.033l rt-- 750 \=- f | f | t jl + 4, 4, | -7© 3.125-21.875 J .00 I (o) -t=.0266 rr 751 * * fl ■{■ fr 4 - 7@ 2.51 =17.57- (b) (a) 16- stringer panel. (b) 15- stringer panel. Figs. 10,11 —1.56 <£. panel -s Figure 10.- Cross sections of test panels. Exact solution Simplified solution ( by three -stringer method) 10 15 20 25 30 Distance from center line of cut-out, in. Figure 1 1. -Stringer stresses in lb-stringer panel with 1 stringer cut and L =8. 3 inches. NASA A 2 4 2 4 2 6 4 cr 40 ID O 2- Exact solution \ c- i.r. 1 1 .. by three-stringer method bimplitied solution 7 y Fig. 13 IC 15 20 25 30 Distance from center line of cut-out, in. 40 Figure 12.- Stringer stresses in lb-stringer panel with 3slring-rscutand L=83 inches. NACA 6 4 a o 4 2 6 4 2 in CO Co 6|- 4 - 2 - Oi- L 6 4 2 4 2 4 2 4 2 Exact solution "\ c . i.r. j ii. } by three- stringer method bimplitied solution] 7 3 Fig. 13 0__ — — — T) " 3 ■ — i s 1 1 1 1 1 10 15 20 25 30 Distance from center line of cut-out, in. 35 40 Figure I3.-Stringer stresses in lb-stringer panel with b springers cut and l_=8.3 inches. N«CA ^ CL m CD i_ Exact solution c- i-r- j i i- \ by three -stringer method oimpiit led solution ; y Fig. 14 15 20 25 30 Distance from center line of cut-out, in. 40 Figure 14.- Stringer s r resses in b-stringer panel with 7stringers cut and L=&3 inches. NACA Exact solution c . ,. r . ... f by three-stringer method bimplit led solution ' Fig. 15 10 15 20 25 30 Distance from center line of cut-out, in. 35 40 Figure I b. -Stringer stresses in 15-stringer panel with 9 stringers cut and L=8.3 inches. NACA Exact solution c . ua i ,. by tnree-stnnger method Dimplified solution Fig. 16 10 15 20 25 30 Distance from center line of cut-out , in , 3b 40 Figure 16." Stringer stresses in lb-stringer panel with 9 stringers cut and L=8.3 inches. Heavy main stringers. NACA 6 4 2 6 4 CL b if) i_ 10 15 ZO 25 30 Distance from center line of cut-out , in . Figure 28.-Shear stresses in 16-stringer panel with 8 stringers cut and L= 1.5 inches. NACA Fig. 29 Exact solution Simplified solution by three- stringer method 10 ^ 15 20 25 3C Distance from center line of cut-out . in. 35 40 Figure 29.- Shear stresses in 16 -stringer panel with 8 stringers cut and L = 8.0 inches. UACA Fig. 30 Exact three- stringer method ■Simplified three-stringer method ■Modified two- stringer method 4 o 2 L 5 10 15 20 lb 30 Distance from center line of cut-out, in. 5b 40 Figure 3QrShear stresses in 16-stringer panel with 8 stringers cut and LH5.0 inches. HACA Fig. 30 Exact three- stringer method -Simplified three-stringer method -Modified two- stringer method i o 2 L -2 2 I c/y Q. CD 5 10 15 20 2b 30 Distance from center line of cut-out, in. 35 40 Figure 3Qr5hear stresses in 16- stringer panel with 8 stringers cut and LH5.0 inches. NACa Fig. 31 Exact solution | c- \r. , i ,. by three-stringer method Simplified solutionl 7 5 10 15 20 25 30 Distance from center line of cut-out , in . 35 40 Figure 31. -Shear stresses in 16-stringer panel with IC stringers cut and LH5.0 inches. NACA Exact solution 1 Simplified solution J b V tHree-stringer method Fig. 32 10 15 20 25 30 35 Distance fronn center line of cut-out, in. 40 Figure 32rShear stresses in 16-stringer panel with 12 stringer scut and LH5.0 inches. mcA IC I Fig. 33 ct3 ,o o 10 Ph •H in II CO . . CO CO to to o CD o 11 rH rd ,0 !-■* o C) O rH rt r! a CI) •H •H ■H 3 lTj o o H, . rH lO LO . o rf rH H co •r! ■0 II II II co iH 1-3 H> 1-1 CD Ph Pi ■p oS +■> +3 -P CO Ph jj pi H I) o o r" o d k +» w u o r-l « «H r-' o --'! o vj •rl +J rf iH o '0 C) « i ;< to <» Fj tO So '< •H <0 -■»■ b r^ t> UNIVERSITY OF FLORIDA 3 1262 08106 503 8 sssggss ss C