I‘Wlxflllm‘l WNW \ \ — — WI | r \l 103—: (DNU1 I_"\J-_‘ LC’AD QiFFU-‘SEQH FRGM AN fiDi’fESEVE-E‘fii‘ifififi STREEQGEER TC‘ AN fiéFEM‘E‘E SHE'E 'Ewsig fc! *i'ne Degma cf M. S. MC‘MGAE‘: $126355 UMWEE‘LL‘EW‘ Lynn C. Laws 1‘93 {$6. THESiS LIBRARY Michigan State University THEE ir .Fugniw..rhfr.fi.lnflvd 53".“ v .‘UIuU fluu‘ Flt ABSTRAC T LOAD DIFFUSION FROM AN ADHESIVE-BONDED STRINGER TO AN INFINITE SHEET by Lynn C. Lewis The case considered here is that of a single stiffener of uniform cross-section and finite length, bonded to an infinite sheet by means of an elastic adhesive. The objective is to investigate the shear stress or shear flow in the adhesive layer, treating all members as linearly elastic. The interaction between the sheet and stiffener is idealized as a line loading. An integral equation in one variable is then formulated for the shear flow in the adhesive layer. This equation is solved in closed form for the case of a rigid sheet, but in general must be solved by numerical means. The solution reveals that the maximum stress concentration occurs at the loaded end of the stringer, with a smaller stress concentration appearing at the Opposite end. These stress con- centrations may be quite large when stiff adhesive layers are used. In this case the load transfer occurs mainly at the ends of the stringer. Thirty solutions have been tabulated together with instructions for inte rpolation. THESI: LOAD DIFFUSION FROM AN ADHESIVE-BONDED STRINGER TO AN INFINITE SHEET BY Lynn C. Lewis A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENC E Department of Metallurgy, Mechanics, and Materials Science 1966 A CKNOW LEDGEMENT I wish to eXpress my gratitude to Professor James L. Lubkin for the guidance that he extended to me throughout this investigation. I am indebted to my brother, Lee B. Lewis, for his many valuable suggestions in the area of computer programming. I am particularly grateful to my wife, Joan, who has been a constant source of encouragement to me throughout my graduate study, for her help in the preparation of the manu- script. Finally I would like to extend my thanks to Mr. Joshua G. Robbins for his help in the preparation of the figures. ii TAB LE OF CONTENTS ACKNOWLEDGEMENT......................... LISTOFTABLES ............................ LISTOFFIGURES............................ NOMENCLATURE............................ I. INTRODUCTION......................... II. MATHEMATICAL FORMULATION ............. III. RESULTSANDDISCUSSION.................. 3.1. Computationalresults.................. 3. 2. Suggestions for further research . . . . . . . . . . REFERENCESOOOO0.00.00...OOOOOOOOOOOOOOOOIO Appendix A. TABULA TED VALUES OF THE ADHESIVE SHEARFLOWOOOOOOOOOOOOOOOOOOOOO Appendix B. NUMERICAL SOLUTION OF THE INTEGRAL EQUATION FOR Q C O C O O C . O O O O O O O O O O O Page ii iv vi 16 16 27 29 3O 40 Table LIST OF TA BLES Dimensionless Shear Dimensionless Shear Dimensionless Shear Dimensionless Shear Dimensionless Shear Dimensionles s Shear Flow Q for B := 31.623 FlowaorB= 10.0. Flow Qfor {3: 3.162 Flowaorfiz 1.0 . Flowaorfiz 0.316 FlowaorB=0.0 . iv Page 31 33 35 36 37 38 LIS T OF FIGUR ES Figure ' Page 2-1 Infinite sheet with finite stiffener attached by meansofanadhesive..................... 4 3-1 ShearflowaorB=31.623................. 2-0 3-2 Shearflowaorfl=10.0.................. 21 3-3 Shearflowaorp=1.0................... 22 3-4 Shearflowaorp=0.316.................. 22 3-5 Shear flow Q for a = 0. 0 and various values of y . . . Z4 |-0 N NHO’UPOHHiID“: :4 NOMENCLATURE width of adhesive layer shear flow in adhesive layer thickness of sheet displacement of stiffener displacement of sheet dimensionless axial coordinate cross-sectional area of stiffener Young's modulus for stiffener Young's modulus for sheet adhesive shear modulus length of stiffener external concentrated load dimensionless shear flow in the adhesive layer axial force in stiffener coordinate directions dimensionless parameters thickness of adhesive layer Poisson's ratio for sheet sheet normal stress in X-direction sheet normal stress in Y-direction sheet shear stress shear stress in adhesive Vi I. IN TR ODUC TION The use of sheet and stringer construction in airframe design has motivated considerable interest in a class of problems dealing with load diffusion from a stiffener into a sheet. This interest has been a result of the demands of the aerOSpace industry for lightweight, high-load- capacity structures. Monolithic construction (sheet with integral stiffener) has been the subject of much investigation. The advent of chemical milling in the aerospace industry has made the monolithic case a very important and practical problem. Buell (1) investigated a semi-infinite sheet with a semi-infinite edge stiffener by means of a complex stress function for the sheet. Benscoter (2) considered an infinite sheet with a finite stif- fener. He observed that the integral equation defining the shear flow between stiffener and sheet was formally identical with Prandtl's equa- tion for the aerodynamic load distribution over a wing of finite Span. This problem was solved approximately, using standard methods for the Prandtl equation. Koiter (3) investigated the case of an infinite sheet with a semi-infinite stiffener. He based his work on Benscoter's equation and presented a rigorous solution based on the application of Mellin transforms. Riveting has also been a common means by which stiffeners have been attached to sheets. Bloom (4) investigated riveted sheet- and-stringer construction using the complex variable method of Muskhelishvili. He found that applying the method of compatible deformations to the sheet and stringer produces an infinite system of coupled algebraic equations in terms of the unknown rivet loads, solvable by truncation. Budiansky and Wu (5) have also contributed to the riveted- stringer load diffusion problem. With the development of better adhesives, a large prOportion of the structural connections in aerospace structures have been accom- plished by adhesive bonding. Not only does the use of adhesives produce more uniform load diffusion in the joint and better fatigue life, but in the case of closed structural systems the adhesive provides adequate sealing for pressurization, fuel reservoirs, etc. The stress distribution in the adhesive layer of cemented lap joints has been investigated by Goland and Reissner (6). Solutions were obtained for two limiting cases, i. e. , where the cement layer has negligible effect on the joint flexibility; and where joint flexibility is mainly due to the adhesive layer. This and other adhesive joint literature is reviewed by Benson (7). In view of the foregoing, the problem which we consider here is of fundamental technical importance in aerospace construction. This is the case of a single stiffener, of constant cross-section and finite length, bonded by means of an adhesive to a sheet of infinite extent. The system is loaded by the application of an external force, acting at one end of the stiffener and in a direction parallel to the longitudinal axis of the stiffener. Our primary objective is to determine the shear flow or shear stress in the adhesive layer in the longitudinal direction. (Benthem (9) has examined the shear flow in a direction transverse to this longitudinal exis, for the case of a bonded hat-shaped stiffener.) In the present problem, the longitudinal shear flow is found to be gov- erned by a one-dimensional integral equation, which appears to be analytically intractable and is therefore solved by a direct numerical procedure, for various combinations of the two dimensionless param- eters governing the problem. Any bending moment introduced by the external force will be neglected, and the sheet, stringer, and adhesive layer are all assumed to be composed of linearly elastic materials. This neglect of bending implies that we are effectively splitting the stiffener into halves, one on each side of the sheet, with each side loaded by half the applied force. The sketches in the following section show the stiffener on one side of the sheet only, however. Having made these assumptions we now proceed with the devel- opment of the governing equations. II. MATHEMATICAL FORMULATION Z Z stiffener :— adhesive ! i \L\ \\\\L\\\\\ i__)_ X 1 1+ Y i t L \infinite sheet ’ ‘ b l" (a) sheet (E2, V) 'r(X) Z .l_4._.c._4_.4__4_1__ 4... m X E— L “—y (M Z A i , stiffener (EPA) P ‘6'“??— —'—'——'——'—"—'—'—' a; X n IY 7(X) (C) Z )l adhesive (G) n W -_-_-_LO9_-_ -_ ._/ 4 \fi\ \\\\\\XXL Y (d) Figure 2-1. Infinite sheet with finite stiffener attached by means of an adhesive. Referring to Figure 2-1, we consider an adhesive layer of small, constant thickness in, situated between a uniform reinforcing rib or stiffener of finite length L and an infinite elastic sheet. The adhesive layer is loaded by a system of continuous shear tractions 7', acting in the Z = 1'] plane and distributed from X = 0 to X = L. These stresses are applied by the stiffener, and their reactions act on the stiffener, both having been induced in the first instance by the external load P. The sheet is likewise loaded by a system of continuous tractions applied by the adhesive layer. These are distributed from X = 0 to X = L in the plane Z = 0. The reactions to these stresses in turn act on the adhesive layer. It is now assumed that the adhesive layer is so thin (reflecting practice) that the adhesive stresses 7' are uniform across the thickness of the adhesive layer. The complex three-dimensional stress disturb- ances at the ends X = 0 and X = L of the adhesive layer are not examined in this study of the overall behavior of the joint. Ad0pting a one-dimens ional theory and neglecting bending, the adhesive shear strain may be expressed in the form Y(X)=[u2(X)-u1(X)]/n . (M) where u2(X) and 111 (X) are the displacements of the sheet and stiffener respectively. The shear stress in the adhesive layer is given by 7(X) = G Y (X) . (2-2) where G is the adhesive shear modulus. Axial equilibrium of a stiffener element of length dX requires that dT (X)/dX + b7 (X) = 0 , (2-3) where T (X) is the axial force in the stiffener and b is the width of the adhesive layer in the Y-direction; see Fig. 2-l(a). The strain in the stiffener is given by the expression dul/dX = T(X)/E1A , (2-4) which integrate s to x u1(X)=(l/E1A) S T(X) dX + u10 , (2-5) 0 where u10 is the rigid-body diSplacement of the stiffener. The sheet diSplacement u‘Z along the line X = 0 may be obtained by means of a routine integration of the known strain distribution in an infinite sheet subjected to a concentrated load acting in the X-direction, together with the principle of superposition. The stress distribution for a concentrated force PO per unit sheet thickness, applied at the origin and acting in the positive X-direction, is given by Timoshenko (8). ox = - (POX/anz) [(v + 3)/2 - (v + l)YZ/r2] (2-6a) «Y = - (POX/anz) [(1/ - 1)/2 + (u + l)Y2/r2] (2-6b) 'TXY : - (POY/anz) [(1 - v)/Z + (l/ +1)X2/r2] (2-6c) Here r is the radial coordinate with respect to the origin: 1/2 r = (X2 + Y2) The normal strains for an arbitrary point are obtained by applying Hooke's law for plane stress, and these are readily integrated to obtain the sheet displacements. Specializing the result to the line X = 0, we get u2(X) = u -[(1+ v) (3 - V)/4TTE2] P In lxl , (2-7) 20 0 where u20 is a rigid-body diSplacement constant. Let the adhesive shear flow q be defined by Q(X) = 7(X) . b - (2-8) Replacing P0 in Equation (2-7)by -q(§)d§/t at x = g and applying the principle of superposition, the sheet displacement for this problem becomes L u2(X) = [(1 + v) (3 - v)/4TrE2t] So q(§)1n|x - gl dg + u20. (2-9) Integrating differential Equation (2-3), and using the boundary condition T(O) = P , we get X T(X) = P - 5 de§ . (2-10) 0 Since q ='rb Equation (2-10) becomes X T(X)=P-S q(§)d§ - (2-11) 0 Since T(L) = 0, we get L p=j gem: . (242) 0 Substituting, Equation (2-11) becomes L T(X) =5 q(€) dé - 5 X q(§) dé , 0 0 which may be written L T(X) = 5X qt) d: . (2-13) Substituting in Equation (2-5) we get X L u1(X) = (l/EIA) So dxl 5 q(x2)dx2 + u10 . (2-14) X1 where x1 and x2 are dummy variables. We can arbitrarily dispose of one of the two rigid-body displacements. Taking u1(0) = 0, Equation (2-9) and (2-14) become X L 111 (X) = (l/ElA) 50 dx1 5 q(x2) dx2 , (2-15a) X l 10 L u2(X) = [(1 + v) (3 - V)/4vE2t] SO q<§>1nlx - gl dg + “20 . (2-15b) Substituting Equations (2-15) into (2-1) and combining this result with Equations (2-2) and (2-8) we obtain an integral equation for q q(X) = B0 - (Gb/ElAn) j X L dx1 S q(x2) de O x l L +[Gb(1+V)(3-V)/4TTE2tnl j q1nlx-eldg , 0 (2-16) where B0 = u20 Gb/n . Making a change of variables to facilitate the use of dimension- less parameters, we define a dimensionless coordinate y and several dummy variables of integration by y=:X/L., yi=)L/L : (i=1,2), u==§/L . A dimensionless shear flow Q and constant C0 are defined by 11 OM = q(x) L/P . C = BOL/P Equation (2-16) becomes y - 1 QM = C0 - (GbLZ/nElA) So dyl Sy Q(Y2) dyz 1 l + [GbL(1 + v)(3 - V)/n4TrEZt] S Q(‘u)[ lnly-ul +ln L] du 0 (2-17) Making the change of variables in Equation (2-12) we see that overall equilibrium requires 1 i 5 Q(y)dy = l . (2-18) 0 Therefore (2-17) may be written 2 Y 1 2 1 QM = K - B So dyl 5 QUIZ) dy2 + Y 50 Q(u)1nly - u l du. y1 (2-19) where K = C0 + y2 ln L , (2-20) 12 and GbLZ/nEIA , (2-213.) .0 ll GbL(l + v) (3 - V)/4TmE2t . (Z-Zlb) .< II The dimensionless parameters {3 and Y are the fundamental physical parameters of this problem. At y = 0, the first integral vanishes and we see that K may also be expressed in terms of 0(0). 1 K = (2(0) - Y2 5‘ Q(u) ln udu . (2-22) 0 Integrating the first integral in Equation (2-19) by parts with respect to yl , and applying Leibnitz's rule, the resulting equation, together with Equation (2-18), completely defines the function Q and the constant K. 2 2 Y _ Q(Y)=K-i3Y+F5 (Y-u)Q(U)du 0 1 + y2 S Q(u) ln ly — ul du (2-23a) o 1 1 = 50 Q(y) dy (2-23b) 13 Equation (2-23a)would be aVolterra integral equation, except for the presence of the'logarithrnic integral. By substituting (2-2321) into (2-23b) we get 2 2 1 Y 1=K-(3/2+(3 S 5 (y-u)Q(u)dudy 0 0 2 - l 1 + y S 5 Q(u) ln ly - ul dudy . (2'24) 0 0 This gives an explicit expression for eliminating K if desired. In the actual determination of Q, however, it proves to be simpler to avoid this procedure. The integral equation with K eliminated is therefore not given here. Consider first the case where y = 0. Equation (2-19) with y = 0 can then be differentiated twice to produce sz/dyz - (320 = o , (2-25) which has the solution . cosh [3y . sinh By + B2 Q=B1 This suggests that we assume Equation (2-19) or (2-23a)to havea solution of the form 14 Q(y) =13;1 sinh By+A cosh By+A3y+A (2-26) 2 4 ° Substituting into Equation (2-23a) with y 2: 0 , carrying out the integra- tion, and equating coefficients of like powers of y, we find Alz-Ktanhfi , A2:K , A3=A4=0 It follows that Q(y) = K (cosh (3y - tanh (3 sinh (3y) . (2-27) Substituting this result into Equation ( 2-23b) and integrating we get K:BCOthB , and thus we have the complete solution in closed form for the case y=0: Q(y) = B coth (3 (cosh (3y - tanh B sinh (3y) . (2-28) 15 This solution could form the basis for a perturbation-type analysis, if y is small. The sheet is effectively rigid when y = 0, if we retain finite adhesive flexibility; see Equation (2-21b). The solution for y = 0 thus reduces to the well-known Volkersen adhesive lap joint theory for the special case of one rigid member; this theory is documented in Benson (7). The solution of Equations (2-23) for nonzero y is obtained numerically here. This is accomplished by dividing the dimensionless adhesive length of unity into N equal increments of length h. The integrals are evaluated numerically, thus generating a system of linear algebraic equations in Q and K. The relation between N and h is given by Nzl/h . The process of integration is carried out by a routine application of the trapezoidal rule, with the exception of the two increments on either side of the logarithmic singularity at y = u. In this region the inte- gration is carried out with Q assumed constant and equal to its value at the singularity. Details of the numerical work are presented in Appendix B. III. RESULTS AND DISCUSSION 3. 1. Computational Results The shear stress distribution in an adhesive layer bonding a finite stringer to an infinite sheet may be computed by a direct numer- ical process. The system of linear equations discussed in Chapter II and Appendix B may be solved by means of a Jordan pivotal-condensation technique. A solution must be obtained for each set of values of the parameters {3 and y. Considering again the two expressions (2-21) defining the nondimensional physical parameters 5 and y we write 5 =(ka/k1)l/2 (3-1a) Y ='(ka/k2) ”2 (3.113) where, ka 2 GLb/n , (3-23) k1 = ElA/L , (3-2b) k2 = 4nE2t/i , (3-2c) l6 l7 and X=(l+v)(3-v)=-V2+2v+3 . (3-3) The parameters ka’ k and k2 are the effective "spring constants" of the 1! adhesive layer, stringer, and sheet respectively. The parameter (3 may then be interpreted as specifying the relative stiffnesses of the adhesive layer and the stringer, while y represents the relative stiffnesses of the adhesive and the sheet. The ratio of ya to (32 may be expressed by : kl/kZ 9 (3'4) and may be interpreted as the relative stiffness of the stringer with respect to that of the sheet. Substituting Equations (3-2) into (3-4) we get 6 = (i/4n) (El/E2) (A/tL) . (3-5) Since the value of Poisson's ratio lies in the interval 0 < Y < 0. 5, it follows that the value of X lies in the interval 3 < R < 3. 75 . For purposes of estimating the size of e, we will assume that X/n z 1. Therefore, Equation (3-5) becomes 18 e = (1/4) (El/E2) (A/tL) . (3-6) In most situations El ’4 E2 , so that the parameter 6 depends primarily upon the geometry of the stringer and the adjacent sheet. Therefore Equation (3 -6) becomes 6 =A/4tL . (3—7) In selecting a range for B, the constants which define B were given values typical for aerospace applications. While some practical problems may have been excluded, it seems likely that the principal phenomena of the problem have been exposed by the present choice. In order to facilitate interpolation in the calculated results, successive values of B increase by a factor of m , so that log B increases by uniform increments. The five values chosen are B = (10)"1/2 , (10)O , (10)1/2, (10)2/2, and (10)3/2 . The parameter B is here regarded as the fundamental one, but only because the case y = 0 (”pure-B problem") offers an exact solution. It appears that y is likely to be comparable to (or smaller than) B in many physical problems, although the exceptional "pure-Y problem" (B = 0) is also conceivable in practice. Once again, the choice of the range of y is thought to expose the characteristic behavior of this parameter sufficiently well for the present purposes. In the present calculations, y appears indirectly: e = \(Z/B2 is used as the primary parameter, and the uniform intervals V? = 0(0. 25)1. 0 are found to 19 give a good spread of the curves Q(y) for any fixed value of B. Thus interpolation in the tabulated results with respect to «I? is facilitated. Note that the size of 6 indicates the importance of the second integral of Equation (2-23a) in the solution. The curves in Figures 3-1, 3-2, 3-3, and 3-4 show represent- ative calculated distributions of dimensionless adhesive shear flow Q, as a function of the dimensionless coordinate y, for various values of B and 6. (Since the curves were plotted by the computer, only Y is available.) The rest of the calculations performed are tabulated in Appendix A. In each figure B is held constant and three curves are given, for the cases V? = 0, 0. 5, and 1. 0. (The corresponding values of y are listed at the t0p of each Figure.) The function Q(y) may be interpreted as the stress concentration distribution in the adhesive since Q: qL/P a (3'8) where P/L is the average shear flow. The shear flow Q may be interpreted in terms of two stressing mechanisms. Referring to the y = 0 curves in Figures 3-1, 3-2, 3-3, and 3-4, we observe the asymmetric stress distribution characteristic of the pure-B mechanism. The curves show that the largest stress concentration is at the loaded end of the stringer (y = 0), and we observe a more-or-less exponential decline as y increases to one. For B > 10, the infinite-stringer case is effectively attained. When B is very large, all load is transferred in the immediate vicinity of the loaded end. 20 N- ”nun Eff-l- 5|.l:a Mn TEE S MEMO 1 Li 1 1 LI | ’ cm 0.0 0.11 0.5 0&0 0.9 0.!0 0.70 0.0 0.50 LE mv Figure 3-l. Shear flow Q for B = 3|.623 21 N- IEDJID ETH- ID.EID Wu lgfi : 7.013 b.0121 I“. '4an .. S-EFPFLCHO ELED .. 7:: _ p j 0.03 0.10 0.5 0.5: on: 0.50 0.00 0.70 0.0: 0.1: .0: MY Figure 3-2. Shear flow Q for 0- (0.0 22 ::: N- 5).!!!) 1113- In!) M- fig 5 3 mm g um. \E! é ; MY Figure 3-3. Shear flow Q for B ' I.O an. N- SLED ETH- 0.5lh m. gm : an. I I G g » LID. E " id 0.” 0.110 0.0 0.10 0.8 0.9 0.0 0.” 0.” 0.70 0.0 0.0 LU WY Figure 3-4. Shear flow Q for“ B . 0.316 ' 23 The pure-B integral equation occurs when y = 0 in Equation (2-23a). By direct differentiation, it may then be reduced to a Volkersen-type differential equation which may be solved exactly, as noted in Chapter II. From this solution, Equation (2-28), we see that the maximum shear flow occurs at y = O and may be expressed in terms of (3. QMAX = Q(O) = (3 coth (3 . (3-9) For (3 > 3 QMAX z ‘3 (3’10) Physically, the pure-B problem occurs when k2 becomes large with respect to ka; i. e., when a flexible stringer is attached to a rigid sheet by means of a flexible adhesive. Representative solutions for the pure-y problem are shown in Figure 3-5. The shear flow distribution here is a symmetric one, with stress concentrations at each end of the joint. This limiting case is achieved when k1 becomes very large in comparison with ka’ as in the case of a rigid stringer attached to a flexible sheet by means of a flexible adhesive. The symmetry of Q(y) in this case may be deduced from physical considerations, by imagining a rigid stringer with loads P/Z at each end, both acting in the negative y-direction. In the case of 24 10— SHEAR FLOW, Q U'l I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 DISTANCE, y Figure 3-5. Shear flow Q for B = 0 and various values of y. 25 large y, the end stress concentrations indicate that load transfer is accomplished at the ends of the stiffener, for the most part. Referring again to Figures 3-1, 3-2, 3-3, and 3-4, or to the integral Equation (2-23a), we see that the geometric parameter E = \(z/B2 controls the admixture of the two stressing mechanisms. As has been pointed out, 6 Specifies the relative stiffnesses of the stringer and sheet. On any one of these fixed-B plots, as 6 increases an adhe- sive stress concentration appears at y = 1 , accompanied by a significant change in the stress concentration at y = 0. This is the effect of adding the y-mechanism. When B is large and the stresses are large, it is physically reasonable that an increase in 6 (or y), i. e. , an increase in the flexibility of the sheet from the rigid state, should cause a decrease in stress concentration. This is observed for B = 103/2 and B = 10. For smaller values of B, the addition of the y-mechanism causes an increase in the stress at y = 0. It is difficult to reason physically about the interplay of mechanisms involved in this phenomenon. The maximum stress concentration occurs at y = 0 in all cases. As 6 approaches unity and the y-mechanism increases in importance, the stress distribution becomes more nearly symmetric with the interior of the joint experiencing a nearly uniform shear flow. As B decreases (more flexible adhesive or shorter stiffener), the magnitudes of the stress concentrations decrease, and in the limiting case when B = y = 0, the shear flow becomes uniformly equal to unity: there is no stress concentration at all. The monolithic case discussed in the Introduction is attained when the stringer becomes an integral part of the sheet, with the adhe- sive layer absent. We can achieve this condition in our present analysis 26 if we multiply both sides of the integral equation (2-23a) by 11/0. Now let the adhesive shear modulus G approach infinity and the thickness 11 of the adhesive layer approach zero. Recalling the definition of e in Equation (3-4), we have y l 5 (y-u)Q(u)du+€S Q(u)ln ly-ul du- y = 0 . (3-11) 0 0 By differentiating this with respect to y, redefining the variables suitably, and letting L » 00 , the present equation can be reduced to that studied by Koiter (3). Since computation has been carried out here only for finite L and finite adhesive flexibility, the two problems are radically different and cannot be compared. The tables in Appendix A present 25 sets of values of the dimensionless shear flow Q as a function of y, for uniform intervals of «f? and log B. This is done to facilitate two-way interpolation by the user. Five-point Lagrangian interpolation with respect to V? and log B is thus possible for at least 51 values of y. Normally, only the values for y = 0 and y = 1 will be of interest in stress analysis. Appendix A also gives five values of the pure-y solution, only three of which are plotted in Figure 3-5. Interpolation here is best done with respect to log \1. While no exhaustive error analysis has been carried out, the results are believed to be reasonably accurate up to B = 10, for all values of e. Basing any conclusions on the established accuracy of the y = 0 family of solutions is admittedly doubtful unless e is of moderate size, unfortunately. This is because the numerical procedure for the 27 y-integral is the one most Open to question. One possible way to test the solution is to fit calculated functions Q(y) accurately with polynomials in y, by sections. The y-integral can then be evaluated analytically and the error in the integral equation for Q(y) assessed. Another way is to increase the number of intervals and examine the results for changes. The sequence N = (number of intervals) = 75, 100, 125 would be illuminating, and would also offer the possibility of a Richardson-type extrapolation for the critical value of primary interest: Q(O). Improved accuracy, in principle, can be achieved for cases of large B and y by using smaller intervals at the ends of the stringer, where the stress gradients are large. In the central regions where the variation of Q is small, a coarser interval is practicable to keep the total number of equations to be solved within reasonable limits. The available time has not permitted these refinements. The purpose of this thesis has been to offer a preliminary investigation of a very complex problem, and it is judged that the principal features of the physical problem have been adequately eXposed. 3. 2. Suggestions for Further Research The principle of superposition allows us to use the solution of this fundamental problem for solving other problems of a more complex nature; for example, a compressive load on one end of the stringer and a tensile load on the other, two end tensile loads, etc. Further research into the distribution of adhesive shear stresses in the transverse or Y-direction (throughout the width of the adhesive layer) should be considered. In the analysis presented here, the shear tractions applied to the sheet are assumed to be continuously distributed 28 along a line coincident with the X-axis (dimensionless y-axis). Line loading is essentially singular in nature as far as sheet behavior is concerned and violates the real physical problem. It is a necessary recourse in a first study, to avoid the complexities of a truly two- dimensional problem. However, the more realistic problem warrants investigation. The effect of bending and the elastic stability of the system should also be considered; bending in particular is of great practical importance. The consideration of the region of influence of the stif- fener in the sheet would be useful for application to problems where several stringers are attached to the same sheet. For example, such a study would reveal when it is necessary to consider stringer inter- action. Another obvious area for further analysis is the finite sheet problem, where the stiffener acts throughout the entire length of the sheet. R EF ER ENC ES E. L. Buell, l'On the distribution of plane stress in a semi- infinite plate with partially stiffened edge, " J. Math. and Phys. , Vol. 26, 1948, p. 223. S. V. Benscoter, "Analysis of a single stiffener on an infinite sheet,‘l J. Applied Mech.,Vol. 16, 1949, p. 242. W. T. Koiter, "On the diffusion of load from a stiffener into a sheet, " Quart. Journ. Mech. and Applied Math., Vol. 8, 1955, p. 164. J. M. Bloom, "The effect of a riveted stringer on the stress in a sheet with a circular cutout, " J. Applied Mech. , 1966. B. Budiansky and T. T. Wu, "Transfer of load to a sheet from a rivet-attached stiffener, " J. Math and Phys” July, 1961. M. Goland and E. Reissner, "The stresses in cemented joints, " J. Applied Mech., 1944, p. A17. N. K. Benson, ”Applied Mechanics Surveys, ” published by ”Applied Mechanics Reviews, ” ASME, 1966. S. Timenshenko and J. N. Goodier, Theory of Elasticity, 2nd ed., McGraw-Hill, 1951, art. 38. J. P. Benthem and J. v. d. Vooren, ”Analysis of panels with bonded, hat-shaped stiffeners, loaded in shear, ” NLL-TN- 5. 520, National Aeronautical Research Institute, Amsterdam, Holland, 1958. 29 Appendix A. TABULATED VALUES OF THE ADHESIVE SHE AR FLOW The following tables give values for dimensionless shear flow, Q = Q(y). Each table provides this information for a specific value of the parameter B and five values of the parameter 6 = YZ/B‘2 . The values of e are invariably taken as follows: «FE: 0, o. 25, 0. 50, 0. 75, 1. o. The values of B are: 10-1/2, 100, 101/2, 102/2, 103/2. In the actual tabulation, the values of Y correSponding to the given values of e are shown as column headings. The variable y is the dimensionless axial coordinate, which ranges from 0 to l, in some cases by intervals of 0. 01 and in others of 0. 02. (Since the tables are printed by computer, only a capital Y is available.) The solution Q(y) is also given when B = 0 ("pure-y" case), for y:10"1/2 100 101/2 102/2 103/2 , , , , . To interpolate, calculate B and y from the physical problem, then form F: y/B and p = 2 (log10 B) -1. For the tabulated values of B, p = -2, -l, 0, l, 2, representing equal intervals. Interpolation with respect to p can then be carried out using Lagrangian interpolation coefficients, up to the five-point formula. Similarly, up to five-point interpolation is possible with respect to V: Interpolation in the pure-y case may be carried out with respect to log10 V. by a scheme similar to that used for B. 30 TABLE 1. GAMMA = Y 0.00000 0.01000 0.02000 0.03000 0.04000 0.05000 0.06000 0.07000 0.08000 0.09000 0.10000 0.11000 0.12000 0.13000 0.14000 0.15000 0.16000 0.17000 0.18000 0.19000 0.20000 0.21000 0.22000 0.23000 0.24000 0.25000 0.26000 0.27000 0.28000 0.29000 0.30000 0.31000 0.32000 0.33000 0.34000 0.35000 0.36000 0.37000 0.38000 0.39000 0.40000 0.41000 0.42000 0.43000 0.44000 0.45000 0.46000 0.47000 0.48000 0.49000 0.50000 DIMENSIONLESS SHEAR FLOW 0 F09 BETA = 31.623 8.74969 5.51104 3.45091 2.45299 1.99644 1.76383 1.61612 1.50307 1.40868 1.32770 1.25767 1.19673 1.14333 1.09616 1.05417 1.01652 0.98255 0.95173 0.92362 0.89788 0.87423 0.85242 0.83224 0.81353 0.79613 0.77993 0.76481 0.75068 0.73746 0.72507 0.71345 0.70254 0.69229 0.68266 0.67362 0.66511 0.65712 0.64961 0.64255 0.63594 0.62974 0.62393 0.61851 0.61345 0.60875 0.60438 0.60036 0.59665 0.59326 0.59018 0.58740 23.717 10.06320 6.34326 3.97269 2.81514 2.27688 1.99707 1.81728 1.67961 1.56504 1.46701 1.38230 1.30859 1.24396 1.18681 1.13588 1.09016 1.04887 1.01135 0.97711 0.94571 0.91681 0.89012 0.86540 0.84243 0.82104 0.80108 0.78242 0.76494 0.74854 0.73314 0.71865 0.70500 0.69214 0.68002 0.66858 0.65777 0.64757 0.63794 0.62884 0.62025 0.61214 0.60448 0.59727 0.59047 0.58408 0.57807 0.57244 0.56717 0.56225 0.55768 0.55344 15.811 13.02620 8.23971 5.17851 3.65494 2.91951 2.52064 2.25880 2.05882 1.89418 1.75459 1.63468 1.53070 1.43974 1.35947 1.28805 1.22406 1.16634 1.11399 1.06628 1.02259 0.98244 0.94540 0.91112 0.87931 0.84971 0.82211 0.79631 0.77215 0.74949 0.72819 0.70816 0.68928 0.67147 0.65465 0.63875 0.62372 0.60948 0.59599 0.58321 0.57110 0.55960 0.54870 0.53836 0.52855 0.51924 0.51042 0.50206 0.49414 0.48665 0.47957 0.47288 7.906 20.72240 13.39070 8.64844 6.11511 4.73624 3.90004 3.32697 2.89734 2.55770 2.28085 2.05073 1.85661 1.69089 1.54795 1.42353 1.31439 1.21799 1.13231 1.05575 0.98699 0.92498 0.86882 0.81777 0.77122 0.72864 0.68958 0.65365 0.62052 0.58991 0.56156 0.53526 0.51081 0.48804 0.46681 0.44699 0.42844 0.41107 0.39478 0.37949 0.36512 0.35161 0.33888 0.32689 0.31558 0.30491 0.29483 0.28531 0.27631 0.26780 0.25975 0.25214 31.623 0.000 31.23480 22.79650 16.63790 12.14310 8.86253 6.46826 4.72082 3.44546 2.51465 1.83530 1.33948 0.97761 0.71350 0.52075 0.38006 0.27739 0.20245 0.14776 0.10784 0.07871 0.05744 0.04192 0.03060 0.02233 0.01630 0.01190 0.00868 0.00634 0.00462 0.00338 0.00246 0.00180 0.00131 0.00096 0.00070 0.00051 0.00037 0.00027 0.00020 0.00014 0.00011 0.00008 0.00006 0.00004 0.00003 0.00002 0.00002 0.00001 0.00001 0.00001 0.00000 TABLE 1. GAMMA = Y 0.51000 0.52000 0 .53000 0.54000 0.55000 0.56000 0.57000 0.58000 0.59000 0.60000 0.61000 0.62000 0.63000 0.64000 0.65000 0.66000 0.67000 0.68000 0.69000 0.70000 0.71000 0.72000 0.73000 0.74000 0.75000 0.76000 0.77000 0.78000 0.79000 0.80000 0.81000 0.82000 0.83000 0.84000 0.85000 0.86000 0.87000 0.88000 0.89000 0.90000 0.91000 0.92000 0.93000 0.94000 0.95000 0.96000 0.97000 0.98000 0.99000 1.00000 (CONTo) 31.623 0.58492 0.58274 0.58086 0.57927 0.57797 0.57697 0.57627 0.57587 0.57577 0.57598 0.57652 0.57738 0.57857 0.58012 0.58202 0.58430 0.58698 0.59007 0.59360 0.59759 0.60208 0.60709 0.61267 0.61886 0.62571 0.63328 0.64164 0.65086 0.66105 0.67230 0.68475 0.69855 0.71389 0.73098 0.75010 0.77159 0.79588 0.82350 0.85514 0.89163 0.93398 0.98337 1.04138 1.11141 1.20386 1.35159 1.64577 2.29490 3.64182 5.76524 32 23.717 0.54953 0.54594 0.54267 0.53972 0.53708 0.53476 0.53275 0.53105 0.52966 0.52860 0.52787 0.52746 0.52739 0.52768 0.52832 0.52934 0.53074 0.53255 0.53479 0.53747 0.54063 0.54430 0.54851 0.55330 0.55872 0.56482 0.57166 0.57932 0.58789 0.59745 0.60813 0.62007 0.63345 0.64846 0.66536 0.68446 0.70615 0.73095 0.75946 0.79249 0.83098 0.87609 0.92941 0.99420 1.08007 1.21639 1.48391 2.06698 3.26987 5.16529 15.811 0.46658 0.46065 0.45508 0.44987 0.44501 0.44048 0.43630 0.43245 0.42893 0.42574 0.42288 0.42036 0.41818 0.41633 0.41484 0.41370 0.41292 0.41253 0.41252 0.41293 0.41376 0.41505 0.41681 0.41909 0.42191 0.42533 0.42938 0.43414 0.43966 0.44603 0.45335 0.46173 0.47132 0.48227 0.49481 0.50919 0.52575 0.54488 0.56714 0.59318 0.62385 0.66024 0.70389 0.75783 0.82990 0.94254 1.15567 1.60542 2.51878 3.95587 7.906 0.24493 0.23812 0.23167 0.22558 0.21982 0.21438 0.20924 0.20440 0.19984 0.19556 0.19154 0.18779 0.18428 0.18103 0.17802 0.17526 0.17274 0.17047 0.16844 0.16667 0.16516 0.16391 0.16294 0.16226 0.16189 0.16184 0.16215 0.16284 0.16394 0.16550 0.16757 0.17022 0.17352 0.17758 0.18250 0.18843 0.19559 0.20420 0.21459 0.22720 0.24262 0.26167 0.28560 0.31655 0.35873 0.42165 0.52768 0.72605 1.10219 1.68883 0.000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -0.00000 0.00000 -0.00000 0.00000 0.00000 -0.00000 -0.00000 -0.00000 0.00000 -0.00000 0.00000 -0.00000 0.00000 -0.00000 0.00000 -0.00000 -0.00000 0.00000 -0.00000 0.00000 0.00000 -0.00000 0.00000 -0.00000 0.00000 -0.00000 -0.00000 -0.00000 0.00000 -0.00000 TABLE 2. GAMMA = Y 0.00000 0.01000 0.02000 0.03000 0.04000 0.05000 0.06000 0.07000 0.08000 0.09000 0.10000 0.11000 0.12000 0.13000 0.14000 0.15000 0.16000 0.17000 0.18000 0.19000 0.20000 0.21000 0.22000 0.23000 0.24000 0.25000 0.26000 0.27000 0.28000 0.29000 0.30000 0.31000 0.32000 0.33000 0.34000 0.35000 0.36000 0.37000 0.38000 0.39000 0.40000 0.41000 0.42000 0.43000 0.44000 0.45000 0.46000 0.47000 0.48000 0.49000 0.50000 DIMFNSIONLESS SHEAR FLOW 0 FOR BETA = 10.000 7.65098 5.04775 3.39134 2.54212 2.10601 1.85383 1.68357 1.55437 1.44982 1.36247 1.28813 1.22401 1.16810 1.11886 1.07512 1.03598 1.00072 0.96877 0.93969 0.91308 0.88866 0.86616 0.84537 0.82610 0.80821 0.79155 0.77602 0.76152 0.74795 0.73524 0.72333 0.71216 0.70167 0.69182 0.68256 0.67387 0.66570 0.65803 0.65083 0.64408 0.63776 0.63184 0.62632 0.62117 0.61638 0.61195 .0.60785 0.60409 0.60065 0.59753 0.59472 7.500 8.09888 5.49772 3.83613 2.94647 2.45476 2.14991 1.93639 1.77317 1.64174 1.53259 1.44008 1.36047 1.29110 1.23003 1.17577 1.12720 1.08342 1.04374 1.00759 0.97450 0.94409 0.91605 0.89010 0.86603 0.84365 0.82278 0.80329 0.78505 0.76795 0.75190 0.73682 0.72264 0.70928 0.69669 0.68482 0.67362 0.66305 0.65308 0.64367 0.63479 0.62641 0.61851 0.61107 0.60407 0.59749 0.59131 0.58552 0.58011 0.57507 0.57039 0.56606 5.000 8.82164 6.34954 4.74805 3.80785 3.21701 2.80928 2.50588 2.26812 2.07518 1.91468 1.77865 1.66161 1.55969 1.47002 1.39044 1.31927 1.25521 1.19722 1.14446 1.09623 1.05198 1.01123 0.97358 0.93869 0.90628 0.87609 0.84791 0.82156 0.79687 0.77370 0.75193 0.73143 0.71213 0.69392 0.67672 0.66047 0.64511 0.63057 0.61681 0.60377 0.59143 0.57973 0.56864 0.55813 0.54819 0.53876 0.52985 0.52142 0.51345 0.50593 0.49885 2.500 9.68315 7.84993 6.56691 5.64236 4.93080 4.35986 3.88944 3.49462 3.15859 2.86943 2.61834 2.39861 2.20507 2.03360 1.88092 1.74435 1.62168 1.51109 1.41105 1.32026 1.23764 1.16224 1.09327 1.03003 0.97191 0.91840 0.86903 0.82340 0.78115 0.74198 0.70560 0.67176 0.64025 0.61086 0.58342 0.55778 0.53378 0.51131 0.49024 0.47046 0.45190 0.43445 0.41804 0.40259 0.38805 0.37436 0.36145 0.34928 0.33781 0.32700 0.31680 10.000 0.000 9.98752 9.03746 8.17777 7.39986 6.69595 6.05900 5.48264 4.96110 4.48918 4.06215 3.67573 3.32608 3.00969 2.72339 2.46433 2.22991 2.01779 1.82585 1.65216 1.49500 1.35279 1.22411 1.10766 1.00230 0.90695 0.82068 0.74261 0.67197 0.60805 0.55021 0.49787 0.45051 0.40766 0.36888 0.33379 0.30204 0.27331 0.24731 0.22378 0.20250 0.18323 0.16580 0.15003 0.13576 0.12285 0.11116 0.10059 0.09102 0.08236 0.07453 0.06744 TABLE 2. GAMMA = Y 0.51000 0.52000 0.53000 0.54000 0.55000 0.56000 0.57000 0.58000 0.59000 0.60000 0.61000 0.62000 0.63000 0.64000 0.65000 0.66000 0.67000 0.68000 0.69000 0.70000 0.71000 0.72000 0.73000 0.74000 0.75000 0.76000 0.77000 0.78000 0.79000 0.80000 0.81000 0.82000 0.83000 0.84000 0.85000 0.86000 0.87000 0.88000 0.89000 0.90000 0.91000 0.92000 0.93000 0.94000 0.95000 0.96000 0.97000 0.98000 0.99000 1.00000 (CONT.) 10.000 0.59222 0.59003 0.58813 0.58654 0.58525 0.58427 0.58359 0.58322 0.58316 0.58342 0.58402 0.58494 0.58621 0.58785 0.58985 0.59224 0.59504 0.59827 0.60195 0.60611 0.61078 0.61600 0.62181 0.62824 0.63537 0.64325 0.65195 0.66155 0.67216 0.68389 0.69687 0.71127 0.72728 0.74515 0.76517 0.78770 0.81320 0.84227 0.87565 0.91434 0.95966 1.01341 1.07831 1.15922 1.26688 1.42811 1.71001 2.26367 3.34945 5.06151 7.500 0.56207 0.55842 0.55510 0.55211 0.54945 0.54712 0.54511 0.54343 0.54209 0.54108 0.54041 0.54008 0.54012 0.54052 0.54130 0.54247 0.54405 0.54606 0.54852 0.55146 0.55490 0.55887 0.56343 0.56860 0.57445 0.58102 0.58839 0.59663 0.60585 0.61614 0.62765 0.64051 0.65494 0.67114 0.68941 0.71010 0.73365 0.76063 0.79178 0.82808 0.87088 0.92207 0.98450 1.06300 1.16704 1.31761 1.56382 2.01442 2.86288 4.19819 34 5.000 0.49219 0.48593 0.48007 0.47460 0.46951 0.46479 0.46045 0.45647 0.45286 0.44961 0.44672 0.44421 0.44207 0.44030 0.43892 0.43794 0.43736 0.43720 0.43749 0.43823 0.43945 0.44118 0.44345 0.44630 0.44978 0.45392 0.45879 0.46446 0.47102 0.47855 0.48716 0.49701 0.50824 0.52107 0.53574 0.55255 0.57190 0.59428 0.62034 0.65094 0.68724 0.73091 0.78434 0.85132 0.93819 1.05680 1.23140 1.51341 1.99990 2.75823 2.500 0.30718 0.29811 0.28957 0.28153 0.27396 0.26684 0.26016 0.25390 0.24804 0.24257 0.23748 0.23276 0.22840 0.22440 0.22075 0.21745 0.21451 0.21191 0.20967 0.20779 0.20629 0.20517 0.20445 0.20414 0.20427 0.20486 0.20595 0.20758 0.20977 0.21260 0.21612 0.22040 0.22552 0.23160 0.23876 0.24713 0.25692 0.26833 0.28164 0.29719 0.31543 0.33692 0.36240 0.39291 0.42989 0.47549 0.53320 0.60940 0.71701 0.87388 0.000 0.06102 0.05522 0.04997 0.04522 0.04092 0.03702 0.03350 0.03032 0.02744 0.02483 0.02247 0.02033 0.01840 0.01665 0.01507 0.01364 0.01234 0.01117 0.01011 0.00916 0.00829 0.00751 0.00680 0.00616 0.00558 0.00506 0.00458 0.00416 0.00377 0.00342 0.00311 0.00283 0.00257 0.00235 0.00214 0.00196 0.00179 0.00165 0.00152 0.00140 0.00130 0.00122 0.00114 0.00108 0.00103 0.00098 0.00095 0.00093 0.00092 0.00091 TABLE 3. GAMMA = Y 0.00000 0.02000 0.04000 0.06000 0.08000 0.10000 0.12000 0.14000 0.16000 0.18000 0.20000 0.22000 0.24000 0.26000 0.28000 0.30000 0.32000 0.34000 0.36000 0.38000 0.40000 0.42000 0.44000 0.46000 0.48000 0.50000 0.52000 0.54000 0.56000 0.58000 0.60000 0.62000 0.64000 0.66000 0.68000 0.70000 0.72000 0.74000 0.76000 0.78000 0.80000 0.82000 0.84000 0.86000 0.88000 0.90000 0.92000 0.94000 0.96000 0.98000 1.00000 DIMENSIONLESS SHEAR FLOW 0 FOR BETA = 3.162 3.81696 2.80297 2.17631 1.80743 1.56959 1.40161 1.27504 1.17549 1.09484 1.02810 0.97196 0.92413 0.88296 0.84725 0.81608 0.78875 0.76472 0.74355 0.72489 0.70848 0.69409 0.68155 0.67071 0.66146 0.65372 0.64742 0.64252 0.63900 0.63687 0.63613 0.63682 0.63902 0.64281 0.64831 0.65568 0.66513 0.67692 0.69139 0.70900 0.73033 0.75618 0.78761 0.82614 0.87396 0.93439 1.01287 1.11903 1.27202 1.51320 1.92877 2.60860 2.372 3.57507 2.78844 2.28668 1.96198 1.73194 1.55782 1.42024 1.30830 1.21528 1.13672 1.06950 1.01141 0.96078 0.91636 0.87717 0.84246 0.81162 0.78416 0.75969 0.73789 0.71849 0.70128 0.68609 0.67278 0.66123 0.65136 0.64310 0.63642 0.63129 0.62770 0.62570 0.62531 0.62660 0.62969 0.63470 0.64180 0.65124 0.66329 0.67834 0.69687 0.71955 0.74723 0.78108 0.82275 0.87462 0.94031 1.02570 1.14132 1.30837 1.57240 1.99480 1.581 3.34636 2.81461 2.45079 2.18331 1.97231 1.79915 1.65355 1.52901 1.42115 1.32681 1.24365 1.16987 1.10407 1.04513 0.99214 0.94438 0.90123 0.86218 0.82680 0.79475 0.76570 0.73942 0.71569 0.69431 0.67514 0.65805 0.64293 0.62971 0.61833 0.60874 0.60092 0.59487 0.59061 0.58818 0.58764 0.58910 0.59267 0.59852 0.60688 0.61801 0.63227 0.65013 0.67218 0.69923 0.73236 0.77313 0.82386 0.88825 0.97313 1.09338 1.27715 0.791 3.20460 2.90578 2.66710 2.46298 2.28302 2.12206 1.97682 1.84499 1.72479 1.61484 1.51400 1.42132 1.33598 1.25730 1.18467 1.11757 1.05553 0.99814 0.94504 0.89590 0.85044 0.80838 0.76951 0.73360 0.70047 0.66996 0.64190 0.61617 0.59266 0.57124 0.55184 0.53438 0.51879 0.50501 0.49300 0.48274 0.47421 0.46740 0.46233 0.45902 0.45752 0.45789 0.46022 0.46465 0.47134 0.48052 0.49253 0.50784 0.52729 0.55263 0.58877 3.162 0.000 3.17206 2.97840 2.79666 2.62610 2.46605 2.31587 2.17494 2.04272 1.91866 1.80229 1.69312 1.59072 1.49469 1.40463 1.32019 1.24104 1.16685 1.09732 1.03219 0.97118 0.91406 0.86060 0.81057 0.76379 0.72007 0.67922 0.64109 0.60553 0.57239 0.54153 0.51285 0.48621 0.46152 0.43868 0.41759 0.39817 0.38035 0.36404 0.34920 0.33574 0.32364 0.31282 0.30326 0.29491 0.28774 0.28172 0.27683 0.27304 0.27035 0.26874 0.26820 36 TABLE 4. DIMENSIONLESS SHEAR FLOW 0 FOR BETA = 1.000 GAMMA = 1.000 0.750 0.500 0.250 0.000 V 0.00000 1.64582 1.50956 1.40346 1.33606 1.31298 0.02000 1.51594 1.42717 1.35569 1.30929 1.29324 0.04000 1.42507 1.36654 1.31767 1.28531 1.27402 0.06000 1.35344 1.31631 1.28431 1.26282 1.25531 0.08000 1.29352 1.27267 1.25407 1.24148 1.23711 0.10000 1.24193 1.23391 1.22623 1.22110 1.21939 0.12000 1.19675 1.19901 1.20040 1.20160 1.20217 0.14000 1.15671 1.16733 1.17629 1.18288 1.18542 0.16000 1.12093 1.13838 1.15370 1.16491 1.16915 0.18000 1.08878 1.11181 1.13248 1.14764 1.15335 0.20000 1.05976 1.08735 1.11250 1.13103 1.13801 0.22000 1.03347 1.06479 1.09369 1.11507 1.12312 0.24000 1.00962 1.04394 1.07594 1.09972 1.10868 0.26000 0.98795 1.02467 1.05920 1.08497 1.09469 0.28000 0.96825 1.00685 1.04341 1.07079 1.08113 0.30000 0.95036 0.99037 1.02852 1.05718 1.06801 0.32000 0.93412 0.97516 1.01449 1.04411 1.05531 0.34000 0.91943 0.96114 1.00128 1.03157 1.04304 0.36000 0.90616 0.94824 0.98886 1.01956 1.03118 0.38000 0.89426 0.93641 0.97720 1.00806 1.01974 0.40000 0.88363 0.92561 0.96628 0.99706 1.00870 0.42000 0.87422 0.91579 0.95608 0.98655 0.99807 0.44000 0.86599 0.90693 0.94658 0.97652 0.98783 0.46000 0.85888 0.89899 0.93776 0.96698 0.97799 0.48000 0.85288 0.89195 0.92961 0.95790 0.96855 0.50000 0.84795 0.88580 0.92212 0.94929 0.95949 0.52000 0.84409 0.88052 0.91528 0.94114 0.95081 0.54000 0.84128 0.87611 0.90909 0.93345 0.94251 0.56000 0.83953 0.87256 0.90353 0.92620 0.93459 0.58000 0.83883 0.86987 0.89861 0.91940 0.92705 0.60000 0.83922 0.86805 0.89432 0.91305 0.91988 0.62000 0.84071 0.86710 0.89067 0.90715 0.91307 0.64000 0.84333 0.86705 0.88766 0.90168 0.90663 0.66000 0.84713 0.86792 0.88529 0.89665 0.90055 0.68000 0.85216 0.86972 0.88358 0.89206 0.89483 0.70000 0.85849 0.87251 0.88254 0.88792 0.88947 0.72000 0.86620 0.87631 0.88218 0.88421 0.88447 0.74000 0.87539 0.88118 0.88252 0.88096 0.87982 0.76000 0.88618 0.88718 0.88358 0.87815 0.87552 0.78000 0.89873 0.89440 0.88540 0.87579 0.87157 0.80000 0.91322 0.90292 0.88801 0.87389 0.86797 0.82000 0.92987 0.91287 0.89145 0.87247 0.86472 0.84000 0.94898 0.92439 0.89578 0.87152 0.86181 0.86000 0.97093 0.93766 0.90108 0.87107 0.85925 0.88000 0.99620 0.95295 0.90744 0.87113 0.85703 0.90000 1.02547 0.97057 0.91497 0.87174 0.85515 0.92000 1.05972 0.99101 0.92387 0.87292 0.85362 0.94000 1.10044 1.01498 0.93441 0.87475 0.85243 0.96000 1.15029 1.04375 0.94706 0.87732 0.85157 0.98000 1.21518 1.08013 0.96286 0.88085 0.85106 1.00000 1.31104 1.13275 0.98547 0.88623 0.85089 TABLE 5. GAMMA = Y 0000000 0002000 0004000 0006000 0.08000 0010000 0012000 0014000 0.16000 0018000 0020000 0.22000 0024000 0026000 0028000 0.30000 0032000 0.34000 0.36000 0.38000 0040000 0042000 0044000 0046000 0048000 0050000 0052000 0.54000 0.56000 0.58000 0060000 0052000 0.64000 0066000 0068000 0.70000 0072000 0074000 0076000 0.78000 0.80000 0082000 0084000 0086000 0088000 0.90000 0092000 0094000 0.96000 0.98000 1.00000 DIMENSIONLESS SHEAR FLOW 0 FOR BETA = 0.316 1008068 1.06867 1.05961 1.05184 1.04490 1.03859 1003281 1002747 1.02251 1001790 1.01361 1.00962 1000589 1.00243 0099921 0.99622 0.99345 0099090 0098856 0098642 0098448 0098274 0098119 0.97982 0097865 0097766 0097685 0097623 0.97580 0.97555 0.97550 0.97563 0.97596 0.97649 0.97722 0.97816 0097932 0098070 0098232 0.98419 0098632 0098872 0.99143 0.99448 0.99789 1.00172 1000604 1.01097 1.01669 1002363 1003339 0.237 1.06011 1005250 1004656 1004134 1.03660 1003223 1002817 1002437 1002081 1001746 1001432 1001135 1000856 1000593 1000346 1.00113 0099896 0099692 0099503 0099326 0099163 0099013 0098876 0098751 0098639 0098539 0098452 0098377 0098315 0098265 0098228 0098203 0098191 0098193 0098208 0098237 0098279 0098337 0098409 0098497 0098602 0098725 0098866 0099028 0099212 0099421 0099659 0099932 1000251 1000639 1001185 00158 1004519 1004071 1003699 1003361 1.03047 1002751 1002471 1002205 1001952 1001710 1001479 1001259 1001049 1000849 1000658 1000475 1000302 1000137 0099981 0099833 0099693 0099561 0099437 0099321 0099213 0099112 0099020 0098935 0098858 0098790 0098728 0098676 0.98630 0.98593 0098565 0098544 0098533 0098529 0098535 0098550 0098575 0098610 0098655 0098711 0098780 0098862 0098959 0099074 0099210 0099380 0099621 0.079 1.03614 1.03354 1.03116 1.02889 1.02671 1.02461 1.02257 1.02061 1.01870 1.01686 1.01507 1.01334 1901167 1001005 1.00848 1.00697 1.00551 1.00410 1.00274 1.00144 1.00018 0.99897 0.99782 0.99671 0099566 0099465 0099369 0099279 0099193 0099112 0.99036 0.98965 0098900 0098839 0.98783 0098732 0098686 0098646 0098611 0.98581 0.98556 0.98537 0.98523 0.98515 0.98513 0.98518 0.98529 0.98547 0.98574 0098612 0998671 0.316 0.000 1003311 1003113 1002919 1002730 1002544 1002362 1002185 1.02012 1.01843 1001677 1001516 1001359 1001206 1001057 1000913 1.00772 1.00635 1.00502 1000373 1000249 1000128 1000011 0099898 0099790 0099685 0099584 0099487 0099395 0099306 0099221 0099140 0099063 0098990 0098921 0098856 0098795 0098738 0098685 0098636 0098590 0098549 0098512 0.98478 0098449 0098423 0098401 0098384 0098370 0098360 0098354 0098352 TABLE 6. GAMMA = Y 0.00000 0.01000 0.02000 0.03000 0.04000 0.05000 0.06000 0.07000 0.08000 0.09000 0.10000 0.11000 0.12000 0.13000 0.14000 0.15000 0.16000 0.17000 0.18000 0.19000 0.20000 0.21000 0.22000 0.23000 0.24000 0.25000 0.26000 0.27000 0.28000 0.29000 0.30000 0.31000 0.32000 0.33000 0.34000 0.35000 0.36000 0.37000 0.38000 0.39000 0.40000 0.41000 0.42000 0.43000 0.44000 0.45000 0.46000 0.47000 0.48000 0.49000 0.50000 DIMENSIONLESS SHEAR FLOW 0 FOR BETA = 31.623 6.63950 4.20694 2.66511 1.92244 1.58600 1.41701 1.31113 1.23075 1.16396 1.10692 1.05781 1.01531 0.97828 0.94577 0.91701 0.89138 0.86841 0.84770 0.82896 0.81192 0.79638 0.78216 0.76911 0.75712 0.74608 0.73590 0.72649 0.71780 0.70976 0.70233 0.69545 0.68910 0.68322 0.67780 0.67281 0.66821 0.66400 0.66015 0.65664 0.65346 0.65060 0.64804 0.64578 0.64380 0.64211 0.64068 0.63952 0.63862 0.63798 0.63760 0.63747 10.000 5.81699 3.86110 2.62210 1.99087 1.66963 1.48585 1.36296 1.27040 1.19596 1.13411 1.08177 1.03688 0.99797 0.96390 0.93384 0.90710 0.88317 0.86163 0.84216 0.82448 0.80837 0.79364 0.78015 0.76775 0.75635 0.74583 0.73613 0.72717 0.71888 0.71123 0.70414 0.69760 0.69156 0.68598 0.68084 0.67612 0.67179 0.66783 0.66423 0.66096 0.65802' 0.65540 0.65307 0.65105 0.64930 0.64784 0.64665 0.64572 0.64507 0.64467 0.64454 3.162 3.16707 2.55542 2.15887 1.90015 1.71613 1.57644 1.46576 1.37544 1.30012 1.23626 1.18136 1.13365 1.09179 1.05478 1.02182 0.99230 0.96573 0.94170 0.91988 0.90000 0.88184 0.86520 0.84992 0.83586 0.82291 0.81095 0.79991 0.78971 0.78027 0.77154 0.76346 0.75599 0.74910 0.74273 0.73687 0.73148 0.72653 0.72201 0.71790 0.71417 0.71081 0.70781 0.70516 0.70284 0.70085 0.69918 0.69782 0.69677 0.69601 0.69557 0.69542 1.000 1.41909 1.35704 1.31255 1.27618 1.24482 1.21707 1.19213 1.16947 1.14873 1.12962 1.11194 1.09551 1.08020 1.06589 1.05250 1.03993 1.02812 1.01702 1.00656 0.99670 0.98741 0.97865 0.97038 0.96257 0.95521 0.94827 0.94172 0.93555 0.92974 0.92428 0.91915 0.91433 0.90982 0.90561 0.90169 0.89804 0.89466 0.89155 0.88868 0.88607 0.88370 0.88158 0.87968 0.87802 0.87659 0.87538 0.87439 0.87362 0.87308 0.87275 0.87264 0.000 0.316 1.04900 1.04332 1.03904 1.03535 1.03204 1.02901 1.02620 1.02358 1.02112 1.01880 1.01660 1.01452 1.01254 1.01066 1.00887 1.00717 1.00554 1.00399 1.00251 1.00109 0.99974 0.99845 0.99722 0.99605 0.99493 0.99387 0.99286 0.99189 0.99098 0.99012 0.98930 0.98852 0.98780 0.98711 0.98647 0.98587 0.98531 0.98479 0.98432 0.98388 0.98348 0.98312 0.98280 0.98252 0.98228 0.98207 0.98190 0.98177 0.98168 0.98162 0.98160 TABLE 6. GAMMA = Y 0.51000 0.52000 0.53000 0.54000 0.55000 0.56000 0.57000 0.58000 0.59000 0.60000 0.61000 0.62000 0.63000 0.64000 0.65000 0.66000 0.67000 0.68000 0.69000 0.70000 0.71000 0.72000 0.73000 0.74000 0.75000 0.76000 0.77000 0.78000 0.79000 0.80000 0.81000 0.82000 0.83000 0.84000 0.85000 0.86000 0.87000 0.88000 0.89000 0.90000 0.91000 0.92000 0.93000 0.94000 0.95000 0.96000 0.97000 0.98000 0.99000 1.00000 (CONT.) 31.623 0.63760 0.63798 0.63862 0.63952 0.64068 0.64211 0.64380 0.64578 0.64804 0.65060 0.65346 0.65664 0.66015 0.66400 0.66821 0.67281 0.67780 0.68322 0.68910 0.69545 0.70233 0.70976 0.71780 0.72649 0.73590 0.74608 0.75712 0.76911 0.78216 0.79638 0.81192 0.82896 0.84770 0.86841 0.89138 0.91701 0.94577 0.97828 1.01531 1.08781 1.10692 1.16396 1.23075 1.31113 1.41701 1.58600 1.92244 2.66511 4.20694 6.63950 10.000 0.64467 0.64507 0.64572 0.64665 0.64784 0.64930 0.65105 0.65307 0.65540 0.65802 0.66096 0.66423 0.66783 0.67179 0.67612 0.68084 0.68598 0.69156 0.69760 0.70414 0.71123 0.71888 0.72717 0.73613 0.74583 0.75635 0.76775 0.78015 0.79364 0.80837 0.82448 0.84216 0.86163 0.88317 0.90710 0.93384 0.96390 0.99797 1.03688 1.08177 1.13411 1.19596 1.27040 1.36296 1.48585 1.66963 1.99087 2.62210 3.86110 5.81699 39 3.162 0.69557 0.69601 0.69677 0.69782 0.69918 0.70085 0.70284 0.70516 0.70781 0.71081 0.71417 0.71790 0.72201 0.72653 0.73148 0.73687 0.74273 0.74910 0.75599 0.76346 0.77154 0.78027 0.78971 0.79991 0.81095 0.82291 0.83586 0.84992 0.86520 0.88184 0.90000 0.91988 0.94170 0.96573 0.99230 1.02182 1.05478 1.09179 1.13365 1.18136 1.23626 1.30012 1.37544 1.46576 1.57644 1.71613 1.90015 2.15887 2.55542 3.16707 1.000 0.87275 0.87308 0.87362 0.87439 0.87538 0.87659 0.87802 0.87968 0.88158 0.88370 0.88607 0.88868 0.89155 0.89466 0.89804 0.90169 0.90561 0.90982 0.91433 0.91915 0.92428 0.92974 0.93555 0.94172 0.94827 0.95521 0.96257 0.97038 0.97865 0.98741 0.99670 1.00656 1.01702 1.02812 1.03993 1.05250 1.06589 1.08020 1.09551 1.11194 1.12962 1.14873 1.16947 1.19213 1.21707 1.24482 1.27618 1.31255 1.35704 1.41909 0.316 0.98162 0.98168 0.98177 0.98190 0.98207 0.98228 0.98252 0.98280 0.98312 0.98348 0.98388 0.98432 0.98479 0.98531 0.98587 0.98647 0.98711 0.98780 0.98852 0.98930 0.99012 0.99098 0.99189 0.99286 0.99387 0.99493 0.99605 0.99722 0.99845 0.99974 1.00109 1.00251 1.00399 1.00554 1.00717 1.00887 1.01066 1.01254 1.01452 1.01660 1.01880 1.02112 1.02358 1.02620 1.02901 1.03204 1.03535 1.03904 1.04332 1.04900 Appendix B. NUMERICAL SOLUTION OF THE INTEGRAL EQUATION FOR Q Writing Equation (2-23a) in the form Q(y) = K - 52y + 62 11m + YZIZW) , (B1) “L where L! y g} 11(Y) = 50 (Y - 11) Q(u) du (32) l 12(y) = S Q(u) ln ly - ul du (B3) 0 we apply the trapezoidal rule for N equal subintervals of length h = l/N. From Equation (B2) we have in general n-l 2 11(yn)=(h/2)[nc20+2>j (n-p)Q] (B4) 9 p=1 where and 40 41 C21 = Q(yi) . To facilitate treatment of the logarithmic singularity write Equation (B3) as the sum of four integrals Iziyn) = I21 + I22 + I23 + I24 ’ where ' Y Y __ n-l _ n 121 - SO qu , I22 — 5 qu , yn-l Y Y _ n+1 _ N 123—3 qu , 124—5 qu , yn yn+1 and Mn, yn) = Q(u) 1n lyn - ul . Note that yN = 1. Since the singularity occurs when u = yn , it seems appropriate to give the greatest weight to the function Q(u) at u = yn . Thus in applying the 42 Theorem of the Mean to this integral we extract the function Q(u) and let Q take on the value Q(yn) = On in the expressmns for 122 and 123. The rest of the integrand is integrable in closed form, with the result that I +1 22 23: ZhQn (lnh- l) . The expressions for 121 and 124 may be treated in the usual manner; in general, the quadrature formula for 12(yn) becomes 12(yn) = ZhQnfln h .. 1) + (h/Z) {QO ln (hn) n-Z +2 2 Qp1n[h(n-p)]+Qn_llnh p=n+2 (135) Special versions of this expression must be written when yn = yo, yl, yN_1, and yN, but these all follow the general pattern described. At N-l +Qn lnh+ 2 Qpln[h(p-n)]+QN1n(1'hn)}- this point we have the capacity to approximate Equation (B1) as a system of N + 1 linear equations in the N + 2 unknowns 00’ Q1, 02’ . . . , QN’ K, by writing Equation (B1) successively at y0 = O, y1 = h, Y2. = 2h, . . yN=Nh=l. ° 9 43 The values of Qn must also satisfy the equilibrium condition (2-23b), which furnishes the necessary (N + 2) th equation 1 1: (O Q(Y)dY , (B6) From Equation (B6), applying the two-point quadrature formula we have N-l 1 = (h/2)QO + h 21 Qp + (h/Z) QN . (B7) p: Combining Ecuation (B1) and (B7), we have a system of linear algebraic equations in on and k, here written with all Specialized forms given in detail: N-l (B1'1)QO+B3QI+Z Vpr+K=O , (B8a) p=2 BZQO+(2B1—I)QI+B3QZ+ Z V p=3 Q 1pp + (VIN/2) QN + K = 52h , (B8b) 44 (UZO/Z) Q0 +(213‘2 + B3) Q1 + (2131 - 1) Oz N-l + B3Q3 + Z VZpr + (VZN/Z) QN p=4 +K = 5211.2 3 (B8c) n-Z 2 (Uno/ )00 + 2 Uanp + (2132 + B3) Qn_1 + (2131 - 1) on p=1 N-l +1330n+1 + 2 VanP + (VnN/Z) QN p=n+2 2 +K=phn (n=3,...N-3) , (B8d) N-4 (UN_Z, 0/2) 00 + 2 UN_2’ pop H2132. + B3) QN_3 p=1 + (231 '1)QN-2 + B3QN-1+(VN-2, N/Z) QN +K=fi2hL(N-2) 9 (B8e) 45 N-3 (UN-1, 0/2) Qo + Z UN-l, pr + (232 + B3)QN-2 p+l 2 +(2Bl-l)QN_1+K = ph(N-1) , (B8f) N-Z (UNO/Z) Qo + 2 UNpr + (232 + B3) QN-l p=1 +(Bl-1)QN+K=BZ.1 , (B8g) N-l (11/2) 00 + h 2 Qp + (h/Z) QN = 1 , (B8h) p=1 where B=Y2h(lnh-l) , 2 B2 Bh/Z . B3 = (vzh 1n h)/2 , 46 Unp = h{)32h(n - p) + Y2 1n [h(n - p) 1} » vnp = vzh 1n [h (p - n)] The computing hardware which is used consists of a Control Data 3600 computer in conjunction with an x-y plotter. The software consists of a program in Fortran IV, and is presented on the following pages. Because of the characteristics of Fortran, it is necessary to index n and p from 1 instead of 0. 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