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'ALADVSOR CO MITEE FORAERONAUTICS TIE.REPORT ORIGOA4ALY ISS"D October 1943 as4 Advance Restricted Report 3JO2 SU=AROMN RECTANGULAR =IOUTS IN MRNSTRINME PANMLS UNIER A=IA LOADS YI ByPaul Kuhn, John 2. Duberg, and Simon 1.Diskin langley. Memorial Aeronautical Laboratory Langley Field,, Va. UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY PR0. BOX 117011 GAINESVILLE, FL 326117011(JM4 WASHINGTON WAIMAM~~~~~~~~~ ~~~~ ThOT r ernso aeroiial sudopoi e rpddsriuino .e ieotr~ reultsto n athoizedgrop rqu ingte foth wa foLT ywrePe vi Ii 9 i1 *!iH i: 1 l 0 (If 3 % o . NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE R ,' T RICTED REPORT STRESSES AROUND RECTANGULAR CUTOUTS IN SKINSTRIHGER PANELS UNDER AXIAL LOADS II By Paul Kuhn, John E. Duberg, and Simon H. Diskin SUMMARY Cutouts 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 cutouts in axially loaded panels. The formulas are derived by means of the substitute strinrger method of shearlag analysis. In a previous paper published under Lhte same title as the present one, the analysis had been basod on a substitute structure containing only tiwo strinrers. 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 Lhreertringer solution as well as of the modified twostrinier solution. In order to check the theory against experimental results, stringer stresses and shear stresses were meas ured around a systematic series of cutouts. In addi tion, the stringer stresses measured in the previous in vestigation were reanalyzed by the new formulas. The threestringer 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 Cutouts in wings or fuselages constitute one of the most troublesome problems confronting the aircraft designer. Because the stress concentrations caused by cutouts are localized, a number of valuable partial solutions of the problem can be obtained by analyzing the behavior, under load, of simple skinstringer 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 cutouts. In reference 2, these suggestions were put into more definite form as a set of formulas for analyzing an axially loaded panel with a cutout (fig. 1). Skinstringer panels, although simpler than complete shells, are highly indeterminate structures. In order to reduce the labor of analyzing such panels, simplifying 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 stitutesingle stringer. In reference 2, this reduction yas carried to the extreme of using only two substitute stringers, one for the cut stringers and one for the uncut stringers, to represent one quadrant of the panel with a cutout. The twostringer structure can be analyzed very rapidly but, being somewhat oversimplified, cannot give an entirely satisfactory picture. In particular, the two stringer structure does not include the region of the net section; and consequently this structure neither shows the effect of length of cutout nor gives a solution for the maximum strihger stresses. These maximum stresses must be obtained by separate assumptions. In addition, there is no obvious relation between the shear stresses in the actual structure and the shear stresses in the substitute twostringer structure as used in reference 2. In order to obtain a more satisfactory basis of analysis than that of reference 2, formulas were developed for a skinstringer structure containing three stringers. At the same time, a new experimental investigation was made. con sisting of strain surveys around a systematic series of cutouts. Stringer strains as well as shear strains in the sheet were measured in these tests, whereas only stringer strains had been measured in most of the tests of reference 2. A study of the threestringer method and of the new experimental results indicated that the accuracy of the twostringer method could be improved by introducing some modifications which have no appreciable effect on the rapidity of the calculations. The main body of the present paper describes the ap plication to a panel with a cutout of a simplified three stringer method of analysis as well as a nodifiedgtwo stringer method. Comparisons are then shown between calculated and experimental results of the new tests and of the tests of reference 2. Appendixes A and B give mathematical details of the exact and of the simplified threestringer methods, respectively. Appendix C gives a numerical example solved by all methods. THEOR3ETICAL ANALYSIS OF CUTOUTS IN AXIALLY LOADED SKINSTRINGER PANELS General Principles and Assumptions The general procedure of analysis is similar to the procedure developed for structures without cutouts (reference 1). The actual sheetstringer structure is replaced by an idealized structure in which the sheet carries only shear. The ability of the sheet to carry normal stresses is taken into account by adding a suitable effective area of sheet to the crosssectional area of each stringer. The idealized structure is then simplified by combining groups of stringers into single stringers, which are termed "substitute stringers"; this substitution is analogous to the use of "phantom members" in truss analysis. The substitute stringers are assumed to be connected by a sheet having the same properties as the actual sheet. The stresses in the substitute sheetstringer structure are calculated by formulas obtained by solving the differential equations governing the problem. (See appendix A.) Finally, the stresses in the actual structure are calculated from the stresses in the substitute structure. It will be assumed that the panel is symmetrical about both axes; the analysis can then be confined to one quadrant. It is furthermore assumed that the crosssectional areas of the stringers and of the sheet do not vary spanwise, that the panel is very long, and that the stringer stresses are uniform at large spanwise distances from the cutout. Symbols and Sign Conventions Al effective crosssectional area of all continuous stringers, exclusive of main stringer bordering cutout, square inches (fig. 2) A2 effective crosssectional area of main continuous stringer bordering cutout, square inches (fig. 2) A3 effective crosssectional area of all discontinuous stringers, square inches (fig. 2) Arib crosssectional area of rib at edge of cutout, square inches (fig. 2) I KL 2 + K2 2 + 2K 1B1 + K12 + 2K n K2 C stressconcentration factor (fig. 7) * CO stressexcess factor for cutout of zero length (Qf"g. 3) D = I + '\ = + 2+ 2 E Young's nodulus of elasticity, kips per square inch G shear modulus, kips per square inch K2 t 2 1 1 72 + X3 1 + A l b1 (T'1 A2) Gt2 Ot2 K3 EblA2 Gt1 K =  4 Eb2 2 K= JKl2K22 _ K3K4 L halflength of cutout, inches (fig. 2) K12 (1 22) 1 i2  p K3K4 P2 = 1 12 X22) P3 =K2 (K,2 12) X\2(%12 2 2) P4 = 34 X2 l2 22) Q1 force Ala acting on stringer 1 at rib, kips Q2 force A2a acting on stringer 2 at rib, kips R stressreduction factor to take care of change in length of cutout (fig. 4) 8 XR difference between actual force in A, (or A2) at the rib and the force Q1 (or Q2), kips  a width of net section, inches (fig. 6) b halfwidth of cutout, inches (fig. 6) bi distance from A2 to centroid of Al (fig. 2) b2 distance from A2 to centroid of A3 (fig. 2) T2Rt2 rl= " T2pt2 TIRti r2 = A2(2R o) Go2R P3 = Eb2T2R tI thickness of continuous panel, inches (fig. 2) t2 thickness of discontinuous panel, inches (fig. 2) x spanwise distances, inches (For origins, see figs. 2 and 6.) y chordwise distances, inches (For origins, see fig. 2.) K2 + K \2 (C12 + K22)2 4J2 21 S=j KI2 + 2 12 + =2 1 4 2 2 V 2 00 average stress in the gross section, kips per square inch 11 stress in continuous substitute stringer, kips per square inch O2 stress in main continuous stringer, kips per square inch 03 stress in discontinuous substitute stringer, kips per square inch rib stress in rib, kips per square inch a average stress in net section, kips per square inch T1 shear stress in continuous substitute panel, hips per square inch T2 shear stress in discontinuous substitute panel, kips per square inch Superscripts on stresses denote forces producing the stresses. Subscript R den:'tes stress occurring at rib station. Tensile stresses in stringers are positive. If the center line of cutout is fixed, positive shear stresses are produced b; a tensile force acting on A1. Simplified ThreeStringer Method A principle for the effective use of substitute stringers was stated in reference 3 substantially as fol lows: 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 cutout, the most important action takes place around the main stringer bounding the cutout. In accordance with the foregoing principle, the threestringer 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 threestringer 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 maxinum 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 the 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 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 RCoA2 0l = A) (1) 02, = o1 + RCO) (2) where the factor Co, for a cutout of zero length, is obtained from figure 3 and the factor R, which corrects Co for length of cutout, is obtained from figure 4. For practical purposes, the parameter B appearing in figure 4 may be assumed to equal unity. (See appendix A.) The length factor R depends, therefore, chiefly on the parameter KIL. This parameter is roughly equal to L/a for usual design proportions; in other words, the length effect can be related nore directly to the lengthwidth ratio L/a of the net section than to the proportions of the cutout itself. The running shear in the continuous panel at the rib station is T 1Rt = 6RCoA2K1 tanh KIL (3) The running shear in the discontinuous panel at the rib station is To 2t = oA 1+ RCo + (4) in which the factor D may be obtained fron figure 5. The stresses a2R and T2R are the maximum values of CO and T2, respectively, and are the raxinurm stresses in the panel. The stress ai reaches its maximum at the center lIne of the cutout. The stress T1 reaches its maximum in the grors section at the station where 01 = 02 t Go0 Stressed in the net section of the substitute structure. The formulas for tie stresses in the net section are ob tained front the exact solution (appendix A). At a dis tance x from the center line of the cutout, the stresses in the continuous stringers are RCoA2 cosh Kx"1 BC A2 cash KIjL Scosh x o2 = ( 1 + PRCo h L As the length of the cutout or, more precisely, the length of the net section increases, the magnitude of the parameter KIL increases and the stresses C0 and 72 converge toward the average stress 0 in the net section. The running shear in the net section is cinh KIx T>t = ORC ArK in1x 1 o 1 cosh K1L and decreases rapidly to zero at the center line of the cutout. Stresses in the gro"s section 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 B) that gives good accuracy in the immediate vicinity of the cutout and reasonable accuracy at larger distances from the cutout. The approximate solution assur.Pes the differ ences between the stresses at the rib station and the average stresses in the gross section to decay exponentially with rateofdecay factors adjusted to give the initial rates of decay of the exact solution. The stress in the cut stringer by the approximate solution is 03 = o(l erlx) (5) The stress in the main stringer is 02 = (o + (02R co)er2x (6) The stress in the continuous stringer 1 follows from statics and is A Al 01 = o + (%co 02) + I (o o) () The running shears in the sheet panels are TltI = Rt erlx 2Rt2 t er2x (8) T 2t2 =. 2t2 er3x (9) Strepsei in the actual structure. By the basic prin ciples of the substitute structure, the stressescin 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 shearlag theory for beams without cutouts (reference 1), the force acting on a substitute stringer is distributed over the corresponding actual stringers on the assumption that the chordwise distribution follows a hyperbolic cosine law. Inspection of the test data for panels with cutouts indicated that neither this nor any 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 nain stringer be assumed to equal ai and that the stresses in all cut stringers be assumed to equal 03. (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 T1 in the substitute structure equal the shear stresses in the first continuous sheet panel adjacent to the rain 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 Ti 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 cutouts. The dis crepancy is of some 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 law of chordwise distribution of the shear stress T2 at the rib station could be approxi mated quite well by a cubic parabola. The effect of this local variation may be assumed to end at a spanwise distance from the rib equal to onefourth 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 72 is distributed according to cubic law, the stress in the rib caused by the shear in the sheet i is orib = r ib1 J (10) Modified TwoStringer Method The twostringer method of analysis given in reference is more rapid than, but not so accurate as, the three stringer nethod previously described. It was found, how ever, that some improvements could be made, partly by in corporating some features of the threestringer method, partly by other modifications. The nain features of the modified twostrinTer 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 shearlag method, however, the stringer substituted for the continuous stringers is located not at the centroid of these stringers but along the edge of the cutout. The substitute structure is used to establish the shearlag paranieter K, which determines the maximum shear stress, the spanwisc 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 twostringer 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 02R = e 1 + 2R(C 1)1 (11) where C is the stressconcentration factor derived in 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 cutouts to 1 for long cutouts, 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 02 = o + (2R o,)eK (12) The stress in the discontinuous substitute stringer is 03 = co(l e'Kx (13) The stress 01 may be obtained by formula (7) when 02 and 03 are known. The running shear in the discontinuous panel is given by T2t2 = oAKeKx (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 threestringer method. 19 EXPERIMENTAL VERIFICATION OF FORMULAS AND COMPARISON OF METHODS Test Specimens and Test Procedure In order to obtain experimental verification for the formulas developed, a large skinstringer 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 cutouts were tested in addition to cutouts of average length, and shear stresses as well as stringer stresses were measured around all cutouts. The general test setup is shown in figure 8. A setup of strain gages is shown in figure 9. The panel was made of 24ST 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 gage 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 loadstress 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. An effective value of Young's modulus of 10.16 x 10 ips per square inch was derived by measuring the strains in all stringers at three stations along the span before the first cutout was nade. This effective modulus may 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 4, percent of the average. The average strain at any one of the three stations in the panel without cutout 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 3 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 0.323. 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 threestringer method. It may be recalled that either method assumes that the stresses in all continuous stringers except the main stringer have the magnitude c1 and in all cut stringers, the magni tude 03. Because the values of oi and 03 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 i.eil v.ith the exferir.ental distribution to be acceptable for most sr;essanalysis purposes in particular, the maIi',.u: :t.rsess in each panel agree fairly well with the calc.uiLarc? ov?es. The most consistent discrepancies are chargeable to the 3implifying assumption that the stringer stresses e.re identical in all the stringers repre sented by one Fubztitute 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 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 ac rather than the actual stresses. On the uncut stringers, however, it may be necessary to allow some extra margin in the stringers near the nain one. Aside from the consistent discrepancies just noted, figures 11 to 32 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 V* 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 threestringer 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 threestringer method are but little larger than the local stress variations that were found experimentally to exist in the panel without cut Co out. Presumably these variations are caused largely by failure of the rivets to enforce integral action of the structure. The errorsin the maximum shear stresses computed by the threestringer method are consistently positive. The discrepancies are possibly caused by the sheet around the most highly loaded rivets 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 cutout to width of panel decreases. The errors given in table 1 for the twostringer method show that this method is decidedly less accurate than the threestringer method for computing maximum stringer stresses but that there is littledifference 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 recalled here that the twostringer 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 wasobtained 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 33, the chordwise variation of the observed and computed rib stresses is shov.'n for three cutouts. Because the chordwise distribution of shear stress in each sheet panel between two stringers is essentially constant, rib stresses computed by formula (C10) will be too small when 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 law 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 lMemorial Aeronautical Laboratory, National dPvisory Committee for Aeronautics, Lai.gJey Field, Va. APPENDIX A TDXACT SOLUTION OP THREESTRINGER STRUCTURES For a twostringer panel constituting one half of a .. symmetrical structure, the application of the basic shear . lag theory yields the differential equation  KT = 0 (A1) which is given in slightly different form in reference 4. In the anal7rris of a skinstringer panel with a cutout, 'a threestringer substitute structure is used. (See fig. 2.) Application of the basic equations of reference 4 to a threestringer structure yields in place of the single equa tion (Al) the simultaneous equations d2T1 2 1.2 T 11 + K3 "T = 0 d2T2 2> (A2)  K.2 T + KT4T 1 0 dx2 J On the simplifying assumption that the panel is very long and that it is uniformly loaded by a stress a at the far ends, the general solution of the equations (A2) is 71 = c le x + c2e 2 (A3) 1___. _lX 1 K 2Xx T2 3 cle +\ 12 72) c2e (A4) in whtch cl and c2 are arbitrary constants. Because the structure is assumed to be symmetrical about the longitudinal as well as the transverse axis, a the analysis may be confined to one quadrant as shown in figure 34(a). The analysis can be simplified somewhat by severing the structure at the rib and considering separately the net section and the gross section. The solutions for the twostringer structure representing the net section can be obtained from reference 4. The solutions for the threestringer structure representing the gross section are obtained conveniently by considering two separate cases of loading. In the first case, stresses aC 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 C necessary to balance the stresses oa. The forces at the rib station ,existing in the stringers are called the Qforces (fig. 34(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 sr.ation. These forces are called Xforces (fig. 34(c)). In the net section the boundary conditions are as follows: At x = 0, TI = 0 (from symmetry) At x = L, XR a  XR 2 A. The substitution of these conditions in the solution of equation (Al) yields X XRK1 sinh Kix T\ coTh KIlT (A5) l d cOsh Y'Ix c" (A6) .. R acoh ]Kx a..= 01 (A7) Ag ccsjn 1iL * The superscript X indicates that the stresses are those caused tby the action of the Xforces. In order to obtain the total stresses, the avrrage stress a must be added to o, or oA. The sheor stress TX is the total stress br caisce the uniform.: stress 0 is not accompanied by any Lhpar stre's. V.n ? he i.1fore c3 uarf Ppplied to the gross section, the bc'.niary c:.!diions at x = 0 are 01 =  C 02 = r2= a 3 = 0 A1 A 2 Applying these conditions to equations (A3) and (A4) gives the following solutions for stresses: 2 (P eklx eA2x) I P2I P4e(A8) (1 x12 1 4K2 22 ) 2 = e e (A9) Y 2 p2 Xlx 4 k (A 10 SQ 1 e2 e / (A10) (2 2K12x12 t \x 4 2 K%_ 2 x 02 t+ 1je tL t1jee 0o A72 1L e71 t] ]1 2 2 l2 1') "lx 4 2, 2x 3 o + A7tI I K3 c 72 K3 The superscript Q indicates that the stresses are those caused by the action of the Qforces. The boundary conditions due to the application of the Xforces are, at x = 0, XR X W1 = 2 = AQ 3 = 0 and the corresponding equations for stress are R Kx k2x1 TX= L(PI + P2)e P3 + P4e j (AlI) r XXRFP, + P2) (c12 q12) X (3 + P4K1l21 22) .2x =2 e T K3 XR I' + P2 1x + P4) .X2x S_ 2 :2x ( + p) ) t2 P1 2 X_ 1,2) X1x 0(P 2 1 1 e)13Xe i AIL ) r 4 22 (xi S P222) If the shear straiis In the net section and in the rngr section, vl>.c' are determined front equations (A5) (".", and (sli), are equated at the rib, the following rel1tii)nrl'.ip ":rvween X^ and Qz results: x' + 7' " "" P+ K1 tanh K (A1) 2 + a q 1F3 FP)r ft r.: ..cr f zerz ecgt.h, L = C, equation (A13) *'X.r, : . 7 2 = C oQ2 V',.. .1 1. 5::,I/ Por any length of cutout, 1 X 1  = QC R NR = Q2C 1 + B tanh KIL =Q2CR where 1/ 2 + K22 + . SKI' + + 2K  Values for Co can be obtained from figure 3. In figure 4, the factor R is plotted for vari.ou. values of K1L and B. The value of B nay be assun.ed equal to unity with little loss in accuracy in the determination of stress; but, if a nore exact solution is desired, the actual value of B nay be computed and the curve in figure 4 cor responding to this value used. APPTEDIX B SIMPLIFIED SOLUTION OF THREESTRINGER STRUCTURES The solutions for the stresses in the gross section given in appendix A are too involved for practical use, and a simplified method was developed. This method assumes that the differences between the values of 02Ro and T2R, obtained by the exact solution, and the corresponding average stresses in the gross section decay exponentially with rateofdecay factors adjusted to give initial rates 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 0a and T1 are then derived from the solutions for 02 and 03. If it is a.simed that the stresses in the cut stringer can be expressed by a = o(l erx) then do3 rIx dx = 0orle but, from the basic shearlag theory, do3 T2 rlx (B dx 3 = orle (B1) Therefore, at x = 0, TR So 2 rl*A'oo The stress in the main continuous' stringer can be approximated by 02 = ao + (a2R ao)e r2x which yields = ,2R o)r2er2x (B2) but, from the shearlag theory, do2 T1t1 T2t2 r2x = + A (02 co) re Therefore, at x = 0, r2 M t2 R 1t) A2(02R cq) The value of a, can be obtained by statics from 02 and a3 and is A2 A 1 = + (o 02) + o 03) (B3) If the value of T2 is assumed to decay exponentially, then T2 T 2er3x and d2 rex dx T2R r3e but, from the shearlag theory, dT2 G "3x S= Eb 2 0 = T2Rr3e Therefore, at x = 0, Go002R G Y R r3 Eb2 T2R The shear stresses in the continuous panel can be determined from the rate of change of 0o. From the shear lag theory, Sdo Tltl (B4) dx A1 Differentiation of formula (B3) yields do1 A3 do3 A2 do2 (B5) dx A! dx AI dx Substitution of the derivatives (Bi) and (B2) already obtained in (55) gives do1 A2 r2x A3 rlx S= 2 o)re x ~A orle (B6) Finally, substitution of (3G) in (B4) yields Tltl = 2R to2e' (T2t2 TlRtl)e r2x APPENDIX C NUMERICAL EXAMPLE Analysis by the Exact ThreeStringer Method The structure chosen for the numerical example is the 16stringer panel tested as part of this investigation. The particular case chosen is the panel with eight stringers cut and with a length of cutout equal to 30 inches. This cutout is the one shown in figure 8. The cross section of the panel is shown in figure 10(a). The basic data are: AI, sq in. . . 0.703 A2, sq in. . . 0.212 A3, sq in. a a . 1.045 tl, in . .. . 0.0331 t2, in. . . .. 0.0331 bl, in . . .. 5.96 b2, in . . 7.56 L, in . . 15.0 These data yield the following values: KI2= 0.01295 K22= 0.00944 K3 = 0.00995 K4 = 0.00785 K = 0.00664 Prom these parameters follow the factors for the rate of decay, which are /12 + 22 + + 2)2 4K2 xl = V 2 = 0.1421 1:1 + K22 VQK2 + K2)2 4K2 X2V : 2 = 0.0467 The computations of stress may more easily be made in terms of the constants PI, P2, P3, and P4, the values of which are K 9(KI 2 2 2) 0.01295(0.01295 0.00218) = 0.0545 1 l 2 X 22 0.1421(0.02021 0.00210) 13 4 P2 = 1i 12" x22) 0.00995 x 0.00785 0.1421(0.02021 0.00210) = 0.0305 12 1 2) 0.01205(O.C1 5 0.0202 1)' 1 3 \(x12 _ 2) 0.0467(0.02021 0.00218) P4 = 2 13 4   0.00995 x 0.00785 0.0467(0.02021 0.00218) = 0.0927 The reduced stressexcess factor is RCO = P4 P2 P1 + P2 P3 4 + Ki tanh KIL 0 00.0927 0.C305 =0.296 0.0545 + 0.0305 + 0.1117 G.0927 + 0.1065 With a force of 7.5 kips acting on the half panel, O = 1.90 = 3.82 kips/sq in. 37 and 0 = .5 8.21 kips/sq in. Therefore, R = RCA.. = 0.296 x 8.21 x 0.212 = 0.514 kip 3 .n 3 _.r: t' . ,'tjon. The shear stress in the substitute cai." c r. , reP , :on is found by equation (A5) ,", : nh i:E1x 1 ; il K, L 05 '.124 x 0.11Z sinh 0.1138x j .0,21 cosh 1.707 = 0.620 sinh 0.1138x At the rib station, x = 15.0 and T1R = 0.620 sinh 1.707 = 1.65 kips/sq in. The stringer stresses are found by substituting in equations (A6) and (A7) and adding the average stress Scaosh K'x =0.514 cosh 0.1138x oI = = E.21  A1 cosh K1L 0.703 cosh 1.707 = 8.21 0.257 cosh 0.1138x R cosh Elx 0.514 cosh 0.1138x 2 A2 cosh K1L 8~ 0.212 cosh 1.707 = 8.21 + 0.850 cosh 0.1139x The maximum stringer stress occurs in the main stringer at the rib, x = 15. The nearest gage location was at x = 13.5, where 02 = 8.21 + 0.850 cosh 1.536 = 8.21 + 2.05 = 10.26 kips/sqin. Stresses in the gross section. The stresses in the gross section are obtained by adding the solutions for the stresses due to the X and Qforces.. The shear stress in the continuous panel is obtained by adding equations (A8) and (A11). The final solution thus obtained is = 0.142lx 0.0467x T e 4.57e At the rib station, x = 0 and T1 = 2.92 4.57 = 1.65 kips/sq in. This value of TI1 checks the one previously obtained for this sane station in the net section. Substituting the constants in equations (A9) and (A10) and cornhininf give T = 2.13e01421x 4.95e0.046?x At x = 1.0O, the point of maximum observed shear stress, To = (2.15)(0.809) (4.95)(0.031) = 6.33 kips/sq in. The stress in the continuous substitute stringer is found by combining equations (A10) and (A12). The final result is 01= 3 82 O.07e0.1421x + 4.63e0.0467x Similarly, the stresses in the main stringer and in the cut stringers are found byadding the proper values of the X and Qstresses. In the main stringer, 02 = 3.32 + 5.55e0.1421x + 1.26e0.467 arnd, in the cut strinaeyS, 03 = 3.32 .46e .14 3.6e Plots of the computed stresses in the panel for this cutout are shown in figures 22 and 30. Analysis by the Approximate ThreeStringer Miethod The basic data are the same as for the exact three stringer method. Compute 21 2 K1 0.01295 x 0.00944 So *= 1.565 '.3fKl4 0.009.5 x 0.00785 0.00664 0.704 K22 0.00944 Prom figure 3, Co = 0.600 From figure 4 for KiL = 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), 0R = 8.21 1 (0.492)(0.600)(0.O' = 7.48 kips/sq in. 02R = 8.211[ + (0.492)(0.600)] = 10.63 kips/sq in. The running shear in the continuous panel at the rib is, by formula (3), TIR t = 8.21 x 0.492 x 0.600 x 0.212 x 0.1138 tanh 1.707 = 0.0547 kip/:.. The maximum running shear in the cut panel is qonputed by formula (4). The value of D is obtained from figure 5; with K = 0.00664 and K12 + Kg2 = 0.02239, D = 0.189 and 0.00785 0.01295 T2 t2 = 8.21 x 0.212 x 0.015 + (0.492)(0.600)+ 0.0664 = 0.234 kip/in. S2 0.234 = 7.08 kips/sq in. R 0.0331 The stresses in the net section are computed as for the exact solution. The rateofdecay factors for the stresses in the gross section can now be computed T2 t2 0.234 r .2 4 = 0.0587 AR30 = 3.82 x 1.045 T2Rt2 T1Rt1 r' 2 p _ A2 2 %o,) 0.234 + 0.055 0.212(10.63 3.82) = 0.1236 G r2R r3 2 _2 b 2T2 0.380 x 10.3 0.07 7.56 x 7.05 The stress in the cut stringers by formula (5) is 03 = Z.82 e e0057x) and in the nain stringer by formula (6) is 02 = 3.82 + 6.93e0.1236x The stress in the continuous stringer can be found by formula (7). The running hears are, by formulas (8) and (9), Tlt1 = 0.234e 00587x + 0.179e0.1218X T,,t = 0.234e00755 At x = 1.50, the point of maximum observed shear stress, T2t2 = 0.234 x 0.893 = 0.209 kip/in. and 0.209 '2 0..0331 = 6.31 kips/sq in. Analysis by the TwoStringer Solution The basic data rernain as before. Compute a 9.38 b ~ 14.06 = 0.667 aA3 1.045 x 9.38 b(A1 + A2) 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) 2R = 8.21[1 + 2(0.492)(0.195)J = 9.G5 kips/sq in. By statics, i,  7.70 kips/sq in. The rateofdecay factor is computed from 2 0.380 x 0.0331 (1 1 \  7.56 0.9715 1.045) = 0.00342 K = 0.0585 In the net section the stringer stresses are assumed to be constant and equal to the stresses at the rib. For the gross section, by formula (12), the stress in the main stringer is 0.0585x 02 = 3.82 + 6.03e and,by formula (13), the stress in the discontinuous stringers is 03 = 3.82(1 e0.0585x The stress in the continuous stringers may be found by using formula (7). The running shear in the cut panel is,by formula (14), T2t2 = 3.82 x 1.045 X 0.0585e0.0585:x = 0.234e0.0585x At x = 1.50, the point of maximum observed shear stress, T2t2 = 0.0234 x 0.916 = 0.214 kip/in. and T_ 0.214 2  0.0331 = 6.47 kips/sq in. 44 REFERENCES 1. Kuhn, Paul, and Chiarito, Patrick T.: Shear Lag in Box Beams Methods of Analysis and Experimental Investi rations. Rep. No. 739, IACA, 1942. 2. Kuhn, Paul, and Moggio, Edwin M.: Stresses around Rectangular CutOuts in SkinStringer Panels under Axial Loads. MNACA A.R.R., June 1942. 3. Kuhn, Paul: Approxinate Stress Analysis of Multi stringer Beans with Shear Deformation of the Flanges. Rep. Io. 636, NACA, 1938. 4. Kuhn, Paul: Stress Analysis of Beams with Shear Defor nation of the Flanges. Rep. Ho. 608, NACA, 1937. tACA r1 i I I I I I 1 45 tC CO   S. cc &3 0 10 Si r4 0U d0 I 0 Ila^ '.4 U C 5 4 ) s s *a 0e 1. r I t. 3*0 S A S0'" U Ca  M o w El A .40 0.. rr .. * 1O M0 a\ W S0 60 4 a rl A 0.4 do CL V .4 wge ^ B0 as . 0* pae Si' * I60 1J cc a :1 4 3< AA rsc o a 1. M 61 1 0. aC i CO3 hoa f) N U.0 go 0 0m SE t. LL M .5 43Me * W  c c C a 2B 4 IN r J a C 0U  M a n n  so so 3* ,C OH IP *!o* 5.4 ue 0* C *gj. mda s m L.e 65 .0 . o~ . .4 00 'a *a9. 9 0 oha au: 1430 1S 5 31u o1 Bo. 0 '0.4% 5' 1 t 1'o at elll l Cl C snIlsiloO I OOCOOOWE 01 o 0 1 0 rw Ll 0n o0mooM . lllslIll o I to0 I I I C 0 ) Cr o4 0 Ch 0 N00000W 01I !I I CW CI *M 10 10 C! CM Io5 10 I Ie I iC 0 H000 0 01 .4.4.4 ..4.4 No r 0850 010 uco CD L, C11 V) 00000000 .4. .... 9.4 r4 H0 'I 0000 03* 10000 0 CNWOCOmcOOCM 101010 10 10 Nr4W4WON  A N 4 .4I .0 * m ulll~ lulIIl Ilull. I51lll llIll IIIlle lull uu lull.' euseIu p10 5O C' 0 C M l I o Oca o0 tI 0 v10o0 I Xt 50 N o to. t I C I 0 10 CM / WN I a 10 *01001.0 lOt CC 0 00o W; ;d) ar01 9 WCDCD CO CD 0 00101Ob0101 00 CW CO CO C "ni. r 0101 .0 ui1 in in io afi H .....0 r4..,4r4 e40 t Ut 0000 qpinc rC4 Cl r4 1 0O01 .4..4. 0000 ' COC 1 01 ciM 00 4 4 0r .4..4. 0000 M 0 C V) .41 I 0000 .401.40' WM ca 00w IN I I 01 i 4C 4 1 L, h MasM 0000 ii m3 "V) 0000 0000 (Mi CM CM C4 0000 5 5 CCO w SI 46 COMPARISON TABLE 2 OF OBSERVED ANID CALCULATED !IAXIMUMII RIB STRESSES [Load on panel, 15 kips] Calculated Number of stringers Halflength Observed Csulated cutout stresses tri/s se Total Cut (in.) (kips/sq in.) (b)in. 16 2 1.5 1.57 2.51 1.48 16 4 1.5 2.20 3.29 1.94 16 6 1.5 2.73 4.42 2.60 16 8 1.5 2.89 5.73 3.58 16 8 8.0 4.30 3.32 2.91. 16 8 15.0 4.77 3.28 2.88 16 10 15.0 5.49 4.48 3.94 16 12 15.0 6.75 6.49 5.70 aFiller strips ineffective. Filler strips effective. Slt I t 1 t Figure 1. Axially l panei Ith cutout. Figure I. Axially luadal panel with cutout. NACA Fi_. 1 NACA Y' Fig. 2 SH Main5tringer I 1I : 7 4 Arib tt / . . ,../ /'/ 5ubshtute structure I, /9"/ / / ,,,? i *1 !I/ Figure i_.5tress distribution around cutout by threestringer method. NACA __ __/ __ __ C o '4 0 'It 02 / o CCC O/ 7/ / . /i/ , Fig. 3 C%o c'j ^ '% gl a) 1 r4 aS LO H Ad 0 .,4 "4 4O 4I I I0 NACA Fig. 4 Fig. 5 +* C L O "4" L I b . ._.' NACA NACA y, ./' .  A4. cutout nj 1W 1W 1W 1W 1W V 1W ~ Lcd &   , 'i Stringer stresses Substitute structure b hear stresses Figure 6.5tress distribution around cutout by modified twostringer method. Fiq. 6 Fig. 7 (1) 0J *NACA ('I black = o10/Z5")  NACA Fig. 8 Ln r4 +3 II 4 V) 4 '4 CO 0 41 UJ (0 I, 11 :(1 t2. NACA Fig.9 w 0 4 O 0 14 p. 0 I IC NACA (a) 6 stringer panel. (b) 15 stringer panel. Figure 10. Cross sections of test panels.  Exact solution  Simplified solution (by three5tringer method) 2  04 2 0 oo~ 10 Distance 15 20 from center line 25 30 of cutout, in. Figure 11 .Stringer stresses in 15stringer ponel with 1 stringer cut ond L =8.3 inches. L 61 4 40  I 40 35 40 Figs. 10,11 ACA Exact solution F Simphf ied 5 olu ion by threestringer method 4 0 ... ... 2 4  4  0 2 n i b 66 0      42 2 4 0   04 _.. 2 0 4 0 0l l 15 I5 3I' ,5 4 Distarce from center line of :jt.ut, in. Figure 12.Sinnger stre5se5 in lstrirer panr,. itf 3 sIlrigrs cut ond L=8.3 inches. BACA Exact solution ,. 1. Simplified solution by threestrirnger rrethod 0 ___ ______ 4 0 6 4 if) b "  4 2 S4    40 04 o4  0 5 10 15 a0 5 30 35 40 Distance from center line of cut)xt, in. Fij re 13.S'ringer stres.e5 in 15stringer panel with 5 ringers cut and L=8.3 inches. Exact solution " impliedd ltir by threestringer method   Simpif'ied solution 2  0 8  40 6  4 e ~ ~ o .. '  . 64 ""  2 ,o) 04 2 4 0  2  D 4L 2 06 0 15 20 25 3 Distance from center lirne c ;ut cjt,irn. 0 3t 40 Figure 14.Stringer stresses in 15stringer panel with 7 stringers c .t ond L=8.31nches. 8 6 F'I 14  2 0 '. to" Ln cu AI 5 NACA Exact solution Fig. Simplified solution by threestringer method  Simplified solution 8 6 o  4 0 101 8 0 00 4 60 0}0 0 4J a ~ ^ 4 0 0 4 _ C, 0 0 5 10 15 20 25 30 35 40 Distance from center line of cutout, in. Figure 15. Stringer stresses in 15stringer panel with 9 stringers cut and L=8.3 inches. Exact solution Exc solution by threestringer method  Simplified solution 6 4 0 0 B 4 2 04 40 08  04 4  4 0 "" 2 04 0 5 10 5 0 5 30 35 40 Distance from center line of cutout, in. Figure 16Stringer stresses in 15stringer ponel with 9 stringers cut and Heavy main stringers. L=.3 inches. IACA Fig. 16 NACA 6 Fig. 17 L' Exact solution o 6 5implified solution by three stringer method 4 6 o 4  2 0 6 6 I  2r 01 I L 6 4 o u u 0 I o^ 0 5 10 15 20 25 30 35 40 Distance from center line of cutout, in. Figure 17. Stringer stresses in 16stringer panel with 2 stringers cut and L=l.5irnches. NACA 6 2F 0 O C 0 8 6 4 0 C. 00  1 r 10 6 4 0~. ___________ 0 ~ I I ~ Fig.ire 18. stringer 5trsses i, 165trinqer panel vwith 4 stringers cut and L=I.l5 inches. 6 F Fii 4 SExact solution S mplified solution by threestringer method  implified solutioni S 18 5 10 l2 20 25 30 Distance from center Iinm of cutout, in. 35 40 ^ EACA Fig. 19 2 80 4 0 8 4 12 0 10 8  :00 a 0 I I I  35 40 Figure 19. Stringer stresses in 16stringer panel with 6 stringers cut and L =1.5 inches. 6 4 0 Exact solution 2  Sified solution by threestringer method 4 0 0 4 2. 0 0 5 10 15 20 E5 30 Distance from center line of cutout, in. I U 0 NACA Fig. 20 mULi oU ogIUI1 5nimplifled solutionI by threestringer method 0  U a a  ~ I I I 35 40 Figure 20.5trinqer stresses in Ibstringer panel with 8 stringers cut and L=l 1.5cne.. 8 6 0 4 St lJ4, 2  08 4 2 6 4 40' 2 0 4: 2 4 0 04 0 0 5 !0 15 20 85 3, Distance from center line of cutout, in. Fi. 21 NACn 8 6 4 2 0 5 10 12 '0 25 3 Diktonce frOn center line ;+ ctcit, in. 35 4,C Figure ?.Stringer stresscs inr I6.rr.ger Danel vvih 8 strinqer cut arind L=. inches E E ,ct solution ,  implified SOt'on by threestringr meth.,j 0 2 12 Lfl Vi', 2 C., 2' .  II  NACA 8 Fg. 6 4  080 6  4" 2 C, 8 ____ cL 0  12 b 8 L 6 + 4 a) I 4 0 4 Exact threestringer method 0 4 5imifed threestrinnger method 2  Modified two strger metho 4 0 0 4 0  _* I  _i / C, 0 D 10 15 20 25 30 Distance from center line of cutout, in. 35 40 Figure 2.5tfringer stresses in 16strirnger porel with 8 .,tringers cut and L=i. inches. NACA 12 Fig. aZ. 8 6 Exact solution 4 rimpliie .t by threestringer method 4 12 0 10 I g8. C) 6 " S 4 4 0 20 4 4 0 , 0 4 "' 2  0 0 4 J  0 ID 15 20 25 30 35 40 Distance from center line of cutout, in. Figure 23.5tringer stresses in 16stringer panel with 10 stringers cut and L= 15.0 inches. NACA 18 16 c 14  10  4 0 12  ,c 4  Exct sution SI 51mphfied scuionJ by three strirer method 60 4 4  04 4 0 40 O  0 0 ', IO 5 ._', 2 ., I, Diar;ce tr rrr :ertEr in, o,'f ,Cut in Figure 4.Stnnger 5reses in 16 r r.]er c ro i 1 strl 'r d Lct :. * _.'r ,. ,. J L O ... FiE. ::A c U) tr) b" u"'" u"; al> L G5 35 40 FI C. 6 p In LI) CL, I 'ft Figure .5.5, e r stresses in 16stringer pinre 2 vit nrir s t rid L=1.5inches.  C1 5lutiun   S[li fie i solt'ln (b. ti ees trnger metl id) o  6 I 0 r  __ I I20 J 30 35 40 Dist n,: frr n r ter i.t in. Figure 6.5her .. ':n 16str'nLur' p.' I,' 4 f, ...r '..ut arJ Lj l.i ncres. 2 0 C 6 . Di: Jr. fr :.rr' :erter lir, , cutour, in Ul) NACA NACA Fig. 27 Exact solution Simplified solution by threesringer method  S im p lifie d so u n 2  2 ,p  U 2'0 c a 4  0 8 L 6 4  4 0 4  04 2 K 0 Figure27Shear stresses in 16stringer parel with 6 stringers cut and L=1.5 inches. HACA Exact solution. xac f solution by threestringer method Simplified solution 2 0 2 00 2 i 4 6r 4 2 4 01 04 2 L o0 5 10 i5 20 25 30 35 40 Distance from center line of cutout, in. Figure 28.5hear stresses in 16stringer panel with 8 stringers cut and L= .5 inches. Fig. 28 NACA 0 0 o  ^r 8 6 4  4 ^ 04 4 0  2  04 0 5 10 15 20 25 30 35 40 Distance from center line of cutout. in. Figure 29. Shear stresses in 16stringer panel with 8 stringers cut and L = 8.0 inches. Fig. 29 Exact solution 1 xc. sid solution by three stringer method  Simplified solutions ri Fig. 30 Exact threestringer method Simplified threestringer method Modified twostringer method p Li U _Z? u 2 0 0 2 0 2 Cn 4 In 4 4 h 04 4 0 2 Distance from center line of cutout, in. Figure 30Shear stresses in 16stringer panel with 8 stringers cut and L=15.0 inches. NACA Fig. 30 Exact threestringer method Simplified threestringer method Modified twostringer method 0 2 2 0 5 .. 0...o 3 5 40 2  .d 4 0 S4 2 2 0 0 5 10 "1 20 25 30 35 40 Distance from center line of cutout, in. Figure 30Shear stresses in 16stringer panel with 8 stringers cut and L=15.0 inches. IACA Fig. 31 Exact solution 5implif led solution by/threestringer method 0 2 0 2 4 8a 6  i i 4 4 4 04 0 ,10 15 D'istanc f'rym " ________ 0 20 25 30 center line jf cutout, in. 35i 4 Figure31.Shear stresses i,. 16stringer panel An !C tringers cut ana L=5.0 inches. NACA 35 40 NA Exact solution \ Simplified solution by threestringer method 4 o o 0 0 , 01 2 S2 S4 S4 6 4 05 0 20 2'5 30 35 40 2 4 Distance from center line of cutout, in. Figure 32Shear stresses in 16stringer panel with 12 stringers cut and L=15.0inches. NIACA I f1  ,I I "I I B .' l C. I U A)  U i i . I L fl. t. LO 01U FI ., .^  . .q I "  .. . .  * ... ..... ''I ,i ,^I ,; I I / ri r r4 ,,i ..: i f !i ___ ,1C I / L _ iI i I r [.. .. J.. .. . I I I / I ! co i c s J N *uL be /S9dJ *ss5J'4g 7 p I ~~L I I I ,~~ j.~p: ~1hItZ _ I~T. TLT~ ___ I ~ I ~ I I I 2. ?ig. 33 0 1 * r.4 (. 14 wI In iw G) ) () 4 ..1 L. t .1 i: 1 ,4  II II II '1 I 0 J .1  .11 SrI r* Q lm t l. L.LI U) c .; c, .,., .,. u. U _______ "~  I.' ~ N I' x I ___ I __ I.  NIACA Fig. '4 F'' .1* 0 1) ~Ij C\. 'I 'C., ~1// .16 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY P.O. BOX 11f7011 GAINESVILLE, FL 326117011 USA *, , '. '1 . 44 11 .' "  
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