- 1 OPTIMUM COLUMN STIFFENING by LUCIEN ANDRE SCHMIT S.B., Massachusetts Institute of Technology (1949) Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE From the Massachusetts Institute of Technology (1950) Signature Redacted Signature of Author Department of Civil and Sanitary Engineering January 14, 1950 Signature Redacted Signature of Professor in Charge of Research................ Signature of Chairman of Department Committee on Graduate Students ..... .*... .090. Signature Redacted ...... e.. ACKNOWLEDGEMENTS The author wishes to express his gratitude to Dr. C. H. Norris, whose patient, considerate and lucid instruction made this thesis possible. The author would further like to take this opportunity to thank Mr. Stanley Falconer and Mr. Harlow Farmer for their helpful suggestions. i TABLE OF CONTENTS Page Introduction B. Purpose C. Scope D. Method E. Results and Discussion F. Conclusions - - - - - - - 1 - - - - - - - - - - - --- A. Preface - - - - - - - INTRODUCTION B. Purpose - - - - - - C. Scope - - - - - - - D. Method - - - ----- INVESTIGATION 10 B. Differential Equations of the Generalized System 11 C. General Solution of the Characteristic Equation 15 D. Optimum Curves for Circular and Square Tubes - 18 E. Minimization Criteria F. Optimum Curves for a Structural Cross Section 27 G. Results and Discussion v - - - - - - - - - - 36 H. Conclusions - - - - - - " - - - - w - - - Formulation of the Prol lem - - - - - - - - - - A. - III. A. - iI. SYNOPSIS - I. 23 42 I ii Page IV. APPENDICES A. Notation B. Discussion of Dr. Dinnik's Paper -- C. (1) - - - - - - - - - - 44 ------ 46 Computations for the Plot which Represents the General Solutions of the Characteristic - - - - - - ---- Equation (13) (2) (3) 51 The Plot of rf-L versus q for k values or ranging from 0.5 to 1.00 - - - - - - - - - Cross Plots of Values of k 4 versus "-- q 64 for Constant -------- 65 cr D. (1) Computations for the Optimum Curves for Circular and Square Tubes (2) .. 82. - from Table II - - o 70 - - - - - - - - - - - --- -- 83 - Computations for Minimization Criteria for Cirocular and Square Tubes- H. - - Computations for Spot Check of Conclusions Drawn - - .----- - - ---a 85 G. - Algebra of Method for Checking Validity of Equation (26) F. - Plot Showing Computed Minimum Points and Average Optimum Curve Drawn E. - -------- - - 86 Computations for Minimization Criteria for Structural Cross Section BIBLIOGRAPHY - - - ---------- --- ------ -- - - - 91- - W M - " - t. 88 91 6 iii LIST OF FIGURES AND TABLES Figure Page 1. Schematic Drawing of Generalized Stiffened System 2. Schematic Drawing of Circular and Square Tubular Cross Sections- 3. Typical Plot of &V 11 - --------------- versus qk 18 for Constant I- 24 cr 4. Schematic Drawing of Structural Cross Section 5. Plot Representing General Solution of the Char- acteristic Equation (13) .6. - -- -- -- -- -- - - 28 --- -- 64 Plot Showing Location of Computed Minimum Points, and Average Optimum Curve Drawn -- - -------- - 82 Table 1. Table of Minimization Criteria G(k,B) 2. Table of Minimization Criteria G(kB) and g(kc,D)-. - - - - - 26 34 1 I. A. SYNOPSIS Introduction The basic ideas for this investigation were stimulated by the common limitations placed on allowable -stresses in Euler range col.umns of constant cross section. If the first step in the design of an Euler range column was to select a cross sectional area sufficient to carry a desired load in direct stress, the second step would be to prevent the column from buckling. The buckling can be prevented by the addition of material which increases the moment of inertia of the column cross section at various points. called stiffening. Such added material is There is an optimum manner in which to distribute the stiffening material for any particular column design. A design method which is more efficient in the use of material than current constant cross section practice can be devised. B. Purpose It is the purpose of this investigation to devise a readily ap.. plicable method for determining how to stiffen most efficiently a column loaded above its initial buckling resistance, against buckling failure. C. Scope This investigation is limited to axially loaded, Euler range columns, with pinned ends. stant cross section. The unstiffened initial column has a con- The initial column and the stiffening are of 2 the same material, and have cross sections with two axes of symmetry. The stiffening used has a constant cross section itself, and is ap*. plied continuously over some fractional length of the column. Fin- ally, it is assumed that the stiffening material acts integrally with the initial column as if there were no interface between them. D. Method For a generalized stiffened column system within the scope of this thesis, a differential equation solution is carried out, which results in a characteristic equation. A general solution of the characteristic equation is developed algebraically and presented graphically. Curves of minimum increase in material solutions, optimum stiffening curves, are developed for circular and square tubu. lar columns, by a method of trial computation. The idea of minimiza- tion criteria is introduced, and optimim stiffening curves for a structural cross section are related tothose already obtained for tubular columns, E. Results and Discussion The graph of the general solution of the characteristic equation of the simple stiffened column system defined has been developed. An average optimum stiffening curve is superimposed on the general solu-4 tion of the characteristic equation. (See Figure 5, Appendix C (3).) For columns which conform to the limitations, and are within the ranges and of the types of cross section considered, the optimum stiffening solution may readily be computed using Figure 5. Although the location of the optimum stiffening curve depends on the shape of the cross section in a strict theoretical sense, the results of this investigation show that the locations of the optimum curves, for various cross sections, do not vary appreciably. troduction of an average optimum curve is reasonable. The in. The average op.* timum stiffening curve presented in Figure 5 may well be applicable to stiffening problems involving far more complicated structural cross sections with tvAo axes of symmetry. F, Conclusions From the results of this investigation, it may be'concluded thatt 1. A readily applicable method for determining the optimum stiffening solution for a column which is within the limitations, and has a cross section of the type, and within the ranges considered, has been dew vised. 2. Although savings of material are achieved by optimum stiffening design, the cost of fabrication probably outweighs any economy of material. In designs where weight is a critical factor, optimum col-. umn stiffening might well find application. 3. The ideas successfully applied to the simplest type of column in this investigation may be applicable to a large number of more com- plicated column types. II. A. INTRODUCTION Preface The basic ideas for this investigation were stimulated by the common limitations placed on allowable stresses in Euler range columns of constant cross section. The reason for such reductions in allowable stresses is that buckling resistance governs the capacity of a long column, rather than resistance to yield in direct stress. As a result of being limited in strength by buckling resistance, none of the material in the column is used efficiently, inasmuch as it is not stressed to its allowable limit. If the order in which the two strength requirements are considered is reversed, a design method which is more efficient in the use of material can be devised. Then the first step in the design of a long column would be to select a cross sectional area sufficient to carry a desired load, stressing the material to the allowable limit. The second step would be to prevent the column from buckling. Assuming there is no reduction in the unsupported length of the col.. umn, buckling can be prevented only by the addition of material to the column. This addition of material which increases the moment of inertia of the column's cross section at various points is called stiffening. B. Purpose It is the purpose of this investigation to devise a readily appe plicable method for determining how to stiffen most efficiently a 5 column loaded above its initial buckling resistance against buckling This problem has two major divisions. failure. 1. What are the possible ways of increasing the buckling re- sistance of a column by stiffening? 2. Which of the possible solutions shows the greatest economy of material? C. Scope Although the concept of column stiffening is applicable to all columns, a thorough investigation of only one very common, but par ticular type of column, has been carried out. The type of column here considered in detail can be described as followst 1. Load: The column is loaded axially without eccentricity and is assumed to be ideal. 2. Ranges (Perfectly straight, homogeneous, and elastic.) The column is in the Euler column range; therefore the material is perfectly elastic and no stresses occur beyond the propor" tional limit of the material considered. If the material is steel$ then this investigation is limited to members having an r ratio equal to or exceeding 100, where L is the length of the column, and r the radius of gyration of the unstiffened column's cross section. More generally, this investigation is limited to columns having an ratio equal to or exceeding TT 4 fL , where E is the modulus ?.L. of elasticity, and fP.L. is the proportional limit stress intensity of the material used, 6 3. End Conditionst The column is pinned at both ends, making it symmetrical about its longitudinal midpoint. A pinned end allows free rotation but no lateral translation at the end of the member, 4. Type of Stiffening: The stiffening used has a constant cross section itself and is applied continuously over some fractional length of the column. It is physically clear that the most effic". ient distribution of the stiffening, for symmetrical end conditions* will be symmetrical about the column's longitudinal midpoint. There.. fore, only symmetrical distributions of the stiffening material are considered. 5, Modulus of Elasticityt It is assumed that the modulus of elas.. ticity of the unstiffened column and that of the stiffening material are the same9 6. Cross Sectiont The column and the stiffening are assumed to have cross sections with two axes of symmetry. 7. Integral Action: It is assumed that the stiffening material acts integrally with the initial column, tween them. as if there were no interface be. The design of a fastening system that will make this assumption a reality is a considerable problem in itself* D. Method The method used in this investigation will now be outlined. An expression relating the buckling resistance of a stiffened column to the amount of stiffening material, and the given properties of the unstiffened column, is derived from the fundamental differential 7 equationt El dy 1 M dx This equation of the beam theory makes the simplifying assump.. tion that d22 [ + (Z)233/2 dx where P is the radius of curvature of the distorted structure. This assumption is sound for small deflections because ( )2 is small com. pared to unity. However, when equation (1) is used in the solution of buckling problems, it leads to a result which implies that buckling failure is a discontinuous phenomenon wherein the lateral deflection instantaneously becomes infinite. The buckling failure of a structure is a continuous physical phenomenon, and a solution based on the exact expression for the radius of curvature would verify this mathematically. See Chapter I, Friedrich Bleich, The Buckling Strength of Metal Struc.. tures. The relation between the amount of stiffening, the given properties of the unstiffened column, and the buckling resistance of the stiffened column, is a transcendental characteristic equation. It can be reasoned physically that the buckling resistance of a stiffened column is equal to some constant, determined by the properties of the unstiffened column, multiplied by a function of the length and bending stiffness (EI) of the stiffening. Physically, then, there are three variables involved, namely the buckling re" sistance of the stiffened column, the length of the stiffening, and the bending stiffness of the stiffening; any two of which may be in.. dependent variables. Certain convenient changes of variable are made 8 in the derivation, but the resulting transcendental characteristic equation contains three variables which are related to the physical variables discussed above. A graph representing a general solution of the characteristic equation in terms of three dimensionless var. iables, which are clearly related to the physical variables, is de. veloped. (See Figure 5.) Given a column with an Euler buckling resistance Pl, it is deo* sired to increase its buckling resistance to Por by stiffening. There are n solutions to this problem, given by any point on the horizontal line dramat fT7 ' is defined as . (See Figure 5.) Note that k or where 1 equals the moment of inertia of 12 the unstiffened cross section, and 12 equals the total moment of inertia of the stiffened cross section. Let q equal the ratio of the stiffened length to the total length of the column. value of of q k to the right of For each k n 0.5, there is a corresponding value which is a solution giving the required increase in buckling resistance. . Values of k to the left of k a 0.5 represent stiffening by values of 12 > 411 and are Lnot considered. The initial column must have an extremely high yR- ratio, to have an optimum stiffenow ing solution for which k<0.5, that is 12 In order to determine which of the each value of jL n available solutions for is a minimum volume solution consider an exoo or pression for AV, the volume of the stiffening material. Let A 1 9 equal the area of the unstiffened cross section, and A2 equal the total area of stiffened cross section. A . AV W (A2 "0 A)1 n ) ( A 1) qIly A2 The relationship between the area ratio - 2 Il = k 12 ertia ratio , and the moment of in" A 1 of the cross section. depends on the shape However, for various cross sections having two axes of symmetry A2 can be expressed in terms of k and certain constants related A1 A2 can to the shape of the unstiffened cross section, Therefore A - be written as h(Kk), where h means a function h of k and Km is a certain constant depending only on the shape of the in. and K m A2 itial unstiffened cross section. AV expression for Substitute h(Kmpk) for in the above. A2 AV q [h(Km k) Having expressed JV. variables the w 1) qLA 1 - [h(Km k) - 1] qAjL is a constant, Since IA of (2 and k and k - 1) q AV in terms of the basic dimensionless this expression and compute AV. n ji-i we substitute or available solutions in q, for any fixed value of values for each of the a minimum value to can be taken as a measure The value of k and AV is called the optimum point. q that gives If such optimum and a curve points are found for a number of fixed values of or is passed through them, it will be a curve of minimum AV solutions. A curve of such minimum volume solutions for any particular initial cross section is called an optimum stiffening curve. (See Figure 5.) 10 III. INVESTIGATION Formulation of the Problem A. Consider an Euler column design for a direct load on a column of length L, pinned at both ends, and with a cross section having two axes of symmetry, cross sectional area A1 , moment of inertia Il, Then Euler's formula gives the buckling resistance of the column as PTI 2EI 2. L criteria. The capacity of the column is limited by the buckling The direct stress intensity P f 1 =A- able stress intensity f m= a SF. is below the allow.- 1 f Y* , where f is the yield point Y? stress intensity of the material used, and S.F. is the safety factor with respect to direct stress intensity. if load by a factor ( Suppose we increase the if ) so that P2 a 1 , then the stress intensity will be up to the allowable limit fa , and we will be making full use of the material. load P2 1 , Of course, the column will buckle under the unless it is stiffened. It would be reasonable to design the stiffened column to carry an ultimate load Por n S.F. P , where S.F. is the safety factor with respect to Vuckling resistance. This is only one rational method of arriv. ing at a value for the ultimate buckling resistance Por to which the column is to be stiffened. Regardless of how P cr is prescribed, or * the problem is how to prevent buckling by stiffening under a load 11 B. Differential Equations of the Generalized System L II Is cL Figure 1. The general fundamental differential equation is 2 EI .0 Yd-x " M 0 0 0 0. 0 0 0 0 (1) dx2 Then over the unstiffened portions (region 1) the following equation is valid: 2 EI d dx Yl PrYl 0 0 (2) 0 0 0 0 * 0 0 *0 Over the stiffened portion (region 2) the following equation is valid 2 d y EI2 dx cr 12 P Let Consider the general solution of equation (2). Then k or equation (2) can be written as 2 d y1 dx + k1 2 y O-0 1 The general solution of equation (2) is ofcos k x + Y 1 1 at2 sin k1x Applying the boundary condition x (4) ......... 0 .,y1 - 0 to equation (4) ofn 0 0 oic 1 +0 1 and Y nsin kx0. 2. . . . . . . . . *1. . (5) P Let Consider the general solution of equation (3). . k - El 2 Then equation (3) can be written as 2 d y2 . 2 -,-+ k 2 y7 2 O dx The general solution of equation (3) is y- cos k x + c sin k 2x . . Applying the second independent boundary condition . . X . . . . (6) 2 Sdy2 to equation (6) 0 I 13 dy2dy M31 -c 0 sn k2 sink2 x 2 x 2 k 2 TL sin k k 2 cos k L 22 L cos kZ tnkL mt tank2 x + o tan k2 L 2 sin k * * * x L Applying the third independent boundary condition yl ' y2 * (7) to equations (5) and (7), c osik sin k, L (1 - q) = o cos k 2 (1 -q) + of tan kL k2 L L sin k2 7 (1 . (8) Applying the fourth independent boundary condition Sdy 2 - x - (1 q) , $ (1 q), 4 cook -ct k sin k2 3 2 2 2 + c4 k 2 co c, 0 and +c' k . ~ to equations (5) and (7), kof cos kL (l.q) - - c k2 sin k (-q) k132 2 -7 1 2 +Ot k + 2 k2L 008k 2ktan*2 * *(9) Divide equation (8) by equation (9) and multiply both sides of the resulting expression by k2 k2 1 L 1 + tan -,- tan k 2 T (1 L w tan k2 E (1 - q) + tan q) k2 L . .. (10) q) 1 14 Employing the trigonometric identity ) tan (a tanA tanC, 1 + tan(C tang equation (10) can be written k2tan k 1 L (1 - q)tan ( ..- k2 or, simplifying, tan k Let k k --2 (l - q) tan k2 L m s.- . . . 0 0 0 0 *. (11) cr a 0 112 Then, replacing k1 by Mr, and k 2 by kPJCr , equation (11) becomes (1 ~ q) tan k k tan if we define a parameter U, such that U q) (1-q) T0 then equation (12) can be expressed as g q 1.. . .. (12) - tan U tan - -. _ U 1.q _ ___ _15 .(13) k where (14) Srq) or solving for Pcr 2U 1 or EI1 2 q L2 Equation (13) is the characteristic equation of the generalized stiffened column system considered. values for k,q, and P or Any set of real positive which satisfy equation (13) constitutes a 6 solution to a specific stiffening problem. This derivation is out. lined by Dr. A. N. Dinnik in his paper on columns of variable cross section. C. (See Appendix B.) General Solution of the Characteristic Equation. The characteristic equation contains three variables, any two of which are independent, the remaining variable being dependent. If q U and are considered to be the independent variables, and k the dependent variable, the problem of transcendental solutions is avoided. If algebraically, U are assigned values, and k q may be computed It will be both convenient and general to plot the general solution of the characteristic equation (13) in terms of the k , - 12 q, and , dimensionless variables or ---------- 16 if 12 is limited to values less than 41I, then the range of k values is 0.5 k a k M 1.00 - k range 0.5 -+* - 1.00 0.500, 0.525, 0.550, k, For each of the ffollowing values of 0.575, 0.600, 0.650, 0.700, 0.750, 0.800, 0.850, 0.900, 0.950; U are assigned, values from zero to - \1 or q, and and are Com.* puted. Use equation (13) to compute q, given k and U 00 * * 0 0* * (13) *a 0 . U q Uta ta Use equation (15) and the expression P to determine the a can be computed, given following equation (16), from which or k and U and q, 2 EI1 ZU or 1 0 q P1 2 ST(1 r2U or L L EI 2 2U T-(1 - q) --- -d) 0 0 0 0 0 0 0 .0 0 0 0 0 (16) 17 (See Appendix C (1).) Sample Computation: U a 0.2000 k a 0.950 Using equation (13) U a tan U tan k 0.2027 tan 1 **q 0.1900 a 1.0526 0.1900. = 5.1928 tan 5.1928 a 1.38055 0.1900 n tan T - 7.2660 7.2660 n 1 + 7.2660 q = 0.8790 T(1 ~q) 2U cr \ V (1 ~q) 2U -if (1 * 0.8790) W~ or r(i , Using equation (16) 2 x a 0.2000 3.14lg 0.9505 0.1210 *1273 0.9505 18 D. Optimum Curves for Circular and Square Tubes For an unstiffened column with a buckling resistance P, it is desired to increase its strength to Por There are n stiffening the column to a buckling resistance P . cr ways of These n solu- tions are given by any point on a horizontal line drawn at or In order to determine which of these (See Figure 5, Appendix C (2).) n solutions is a minimum volume solution, an expression for terms of k and q AV in must be developedt A V Since IA - (A2 A 211 ) Lq - (). 1 )qL . .. * . 1 is a constant, it remains only to express (17) as a function AW of k or q. The relationship between V- and k depends 2 on the cross section shape, A The derivation of the expression for - 2 in terms of A1 now be carried out. -4 - C11 C(I I 'I CIO K K Figure 2. k will 19 For circular tubes, For square tubest I, I 2 I (d24 P d T I A1 = d 2 0~L~ 14 A2 to k To relate let d t q - let B F12 40 1 d .d d and note that 0 t B 2(B + 1) Then for circular and square tubes 4 (d ( 4 . do) 4 4 (d 2 m do I1 - k 2 4 4 2 d 4 1 d 2 B - d or 0 Therefore* I 0 0 * But (d 4 # d 4) 1 a 1 d (d 24 2 12 2 A2 nd2 2 *d d o (d2 A2 - 1 12 2 o d ) (d ) ow d 0 2 .. d 4 ) A, 64 (d1 d 0 -4- 1+ B 0 0 20 4 4 d d1 (1 + B) k2 4 d (1 2 + B) 4 1 (1+ B) d2 2 d2 22 1 (1 + B) 4 d d 1- (1 + B) 1 ( (1 + B) k2 . k ( 1 (1 + B) k (1 + B) 4 (1 + B)4 " 1 + k2 k- (1 + B)4 (1 + B)4 - 1 + k d k(1 +B) 4 But 2 A2 17 A1 d0 (d 22 a d 2 1 A2 A ( (1+ B) 1 A2 _ 1/k 2 + k 4 (1+B) d_2 k (1 + B)__ d 2) (d 42 d1 (1 + B) d (1 + B) kl + B)4 k(1 + B) 2 (1 + B) 21 (1 + B)4 1 + k2 1 0 A B + 2B 0 0 * 0 * (18) 21 A Substitute for from equation (18) into equation (17): AV k . B q LA1 . . (19) + 2B Equation (19) is an expression for AV in terms of the variables k and q. Recall that for each particular value of we had a set or of as n solutions in terms of d d0 q and k. The constant B defined is a convenient constant which depends on the 0 t of the unstiffened column, lower limit of t 1 If an upper limit of - ratio d1 1/3, and a is adopted for consideration, the range of practical tubular columns is completely covered. These limits were adopted after a study of available tubing as indicated by various handbooks. See p. 15, Handbook of Welded Steel Tubing, Formed Steel Tube Institute, 1621 Euclid Avenue, Cleveland, Ohio. January 15, 1941. See Section I - 16, Summerill Aircraft Tubing Data, Summerill Tubing Company, Bridgeport, Conn. March 8, 1943. Localized buckling failure of the unstiffened portion of the tube is not critical. Since this work is limited to the Euler range the stress intensity in the unstiffened portion of the column will never exceed the proportional limit. See p. 440, Theory of Elastic Stability, Timoshenko. 2Et cr for steel f 30,000 d 2 E = 30 x 106 -#/in2/r 22 t 30 x 103 a 2 x 30 x 106 3(1 - 1/9) d t 0.8165 x 10' r - - 100 t ( Since such thin tubes 1 ) 10 are not practical, it is clear that localized buckling of the unstiffened portion of the tube is not critical for the columns considered. L For , B - 2 and the expression for AV (equation (19)) becomes k V80 + kT AV 1 - 81 .J q LA1 . . . . . (20) - 0.975, 0.950, 0.925, 0.900, 0.875, 0.850, 0.825, 0.800, For or 0.775, 0.750, 0.725, 0,700, 0.675, 0.650, 0.625, 0.600, 0.575, 0.550; a cross plot of k is drawn. q versus (Appendix C (3).) The ac- curacy of these cross plots depends on the number of constant k curves, and is the chief source of computational error in this work. Then for each value of the k , q solution which makes &V or a minimum is found by trial, using the cross plot and equation (20). (See Appendix D (1).) These minimum volume solutions are plotted on the graph represent" ing the general solution of the characteristic equation. A smooth curve joining them is the optimum stiffening curve for circular and t square tubular columns with a For 7- a (equation (19)) 17 , becomes B 11 1 and the expression for AV 23 &y . 0406 + k 21 0.0201 [/k 1 - 1] q LA The procedure is identical with that outlined for that equation (21) must be used to compute AV. .. t 1 - 0 d1 (21) . except (See Appendix D (1).) The resulting minimum volume solutions are plotted on the graph representing the general solution of the characteristic equation. A smooth curve joining them is the optimum stiffening curve for cir- t 1 t . cular and square columns with a Due to the lack of accuracy in the cross plots of k versus q for fixed 4J! values, the points representing minimum volume solu. or P tions do not lie in the smooth curve. For each value, the or curve of AV versus k,q solutions is quite flat, near the minimum point fcr t AV. The minimum points making up the optimum curve for 1 Salmost the same as those for 1 t 1 - It has there* 1 fore been decided to pass one. average curve through the two sets of minimum points, which will be the optimum stiffening curve for cir.- t1 cular and square tubular columns with a between 1/3 and 0 (See Figure 6, Appendix D (2).) E. Minimization Criteria. For the square and circular tubular columns considered, the If the expression for . l /k t a- 1 and t 1 - &V, equation (19), is consideredj (1+ B) 4 B + 2B 1 + k2 1 ]LA . optimum curve is essentially the same for 24 recalling that for t 1 -d 1 31 B = 2, t I d and for B 2 1 100 it might be suspected from the form of equation (19) that the kq solution which makes AV a minimum is almost independent of B or t d1 It can be concluded from the computational work that has only one maxima-minim& solution between . at each line of constant q - 0 and A typical curve of k,q solutions at any line of constant AV q = 1.00, &V versus is shown in Figure 3. or ASYMPTOTIC cnV -1 0 1.00 % ASSOCIATED Figure WITH A 3. Since there is only one real maxima..minima solution, d be the criteria for this one solution. - o must It must be established that 25 0 - is the criteria for the one real minimum solution by the reasoning above, because q n f(k) for each line of constant cr is not known, except graphically. If f(k) a q could be expressed explicitly, then the conventional second derivative test could be used to determine which of the solutions of d dic 0 was the real minimum point, and also to show that this was the only real minimum point. Since dk - 0 is the unique minimization criteria, the exact form of this criteria will now be determined. with q replaced by V f(k). 1/k (f + B) 4 - 1 + k2 B [1/k V(1 + B)4 -1 AV - Consider equation (19) o 1 + 2B + k2 1 - J B2 + 2B)1 f(k) LI 2 iaU B +2B dAV (+B) 4 1 + k2 1/2+[(+B )4 Ik + Let dAV 0 + k2 ]/21 +2B 1)B 1/k [(l+B)4 " 1 + k2]1/200 1 - (B and multiply both sides by f (k) ) B + 2B k 2 [(+B)4w 1 + k2 1 t(l+B) 4 0 + 1+ kc2 1 + k 23 f k + Then (ic((l+B)4..+kcJ*(B+l)2kc2(l+B)4.4+i2l /3 B Z +2 26 ivhiph can be reduced to f(k) = G(kB) fr(k) . . . . . .0 0 0 0 (22) where G(kB) - k(+B) 4 1 +k2-(B+) 2 22 [(4+B)4-+k2 1/2) 1 (1+B) . . . .(22a) Equation (22) says that at any line of constant where or q a f(k) is given graphically by the cross plot, the value of k such that P(k) f(k) G(kB) is satisfied, is the minimum volume solution. Each value of k from 0.5 to 1.00 is a minimum volume solution at the appropriate value of . If for k n 0.5, 0.7, 0.9, 1.00, or G(kB) is computed with B = 2 and then with B = , using equation (22a), the results can be tabulated as follows. (See Appendix G.) Table of Minimization Criteria G(kB) 1 &G(kB) 100 G(kB)ave 1.00 0 0.90 0.08995 0*08573 4.8 0.70 0,20972 0,18022 15.1 0.50 0.24961 0.19267 25.75% % % 0 Table 1. A rough check on the G(k,B) values computed can be made. For a value of k, say k 0.7, from the optimum stiffening curveppit can be seen that k a 0.7 is a minimum volume solution between n 0.725 cr VL 27 and 0.750. Draw tangents to the cross plots for 0.725 , 0.750 ; or at the q value associated with k = 0.7 on the optimum curve, namely q = 0.54. Then q Ak & q 1''"kJdq G(kB) a f dk G(kB) = q k 0.59 x 0.20 .020 The fact that this rough graphic check falls between the two G(kB) values computed indicates that everything fits together. 1 that is, as k Table 1 shows that for lower values of , or decreases, the minimization criteria for B a 2 and B increasingly different. - become However, for the f(k) functions involved3 a difference of 25.75% in the minimization criteria G(kB) doesn't change the location of the optimum stiffening curve on the graph rep. resenting a general solution of the characteristic equation appreci.- ably. F. (See Figure 6, Appendix D (2).) Optimum Curves for a Structural Cross Section. The general solution of the characteristic equation is in- dependent of the shape of the cross section. plots of k versus q Therefore, the cross for various constant values of \f-i1 are or still valid. Then for a structural cross section using equation A (17), it is only necessary to express as a function of k. - I 28 NOTATION. t Kb t Figure 4. The cross section that is to be considered now is made up of four angles arranged at the corners of a square and stiffened over a leaph Lq by four plates. Over the unstiffened portion of the column, the four angles would be held together by lacing bars. With A the notation indicated in Figure 4, an expression for in terms of k will be dorived, g(d --2b) t 1l4 t (d"2) I but t and do ~(d dc d1 -. 2t 4 -d, )-"2 (d -2b)t( b n tD n cDd (1 (l - 2c) c)d d d 1 +d 2 1 P) 2 3 1 3 29 (1 - 24 -2D 2(1*20D) (1-2c) 2 3 c 1 + (1-2c) 1 2 c + Tim 2 3) 4 El s (1-2e)4],m6c(l-2eD)(1-c)2-2(1--2cD)c'3 'a 1 "I 2c (1 - 2oD)3] d14 11 - Gd1 4 *... .0 ....... . . (23) where G- f[lo(1-2c)4]-6c(1-2cD)(1_c)2 -2(l.2cD)c3. 2c(1..QcD)13 . . . . . I is computed as IS + I (d2 4 - d 4 S is the moment ,where ) )23a) of inertia of the cross hatched corners in Figure 4. These corners can not actually be provided if the stiffening plates are to be welded to the angles. assumed to be zero. However, in this derivation, the term G is This approximation is on the unsafe side, and will be considered to trespass slightly on the safety factor. 1 12 = 1l + A I = Gd1 4 + Tr 0 assumed approximation. I2 tG + 212 L ( d1 4 -- d 4)u Therefore )4 0 .. 12 I d 1 . . *0 From the definition of k, and equations (23) and (24) , 6- 4 * (24) --i 30 k 1 2 G .- [G + 1 A1 w t1 - (1 A, 4c A2 A - d 20) 2 [22D - 11 d u1J+1 . . . . . (25) - 2b) _ 4c(1 - 2cD)] d1I 2 2 &A [8Dc A2 d A2 A d1 2 [8Dc 1 d 2 2 - d 2 -. 4c ) + (d2 2 2 12G 4t (d sd 2 A A2 = A 3 + 1 k d2 - d 2y 1 k 1 + G t41 12G d1 1 TE 4 d ) d2 )4 - 12 1 T2- 2 - 4e2 2 11] + d2 1] + d 4c (2D-1) Using equation (25) to eliminate d2 from the expression for A2, A1 Then -8Dc2 .s 4c2 .. 1 4c (2D 00 V12G 1C 2 . 1] 1) + 2 + 1 1 31 A2J + H V12G[ 1J + 1 2 . . . . . (26) . . . (26a) 0 0 * 0 . (26b) where 2 2 S8D2 (4c .. 1 2J " 402 (2D -d 1) 1 - i . . and H n1 0 . 0 0 0 4c (2D - 1) This is the desired relation between section under consideration. A A for the cross and k In order to check this expression for A2 d in terms of k, consider the limiting case when b , that is, the initial cross section of four angles is a closed square, If for this case we relate c and D to the constant B, using their respective d1 definitions and the special condition that b , (equation (26)) should reduce to equation (18). This has been done, and since equation (26) did reduce to equation (18), it constitutes a check of sorts on the validity of equation (26). An expression for (See Appendix E.) AV in terms of k and q can be developed as follows: A2 AV- ( - A11 1 ) q L A1 0 . 0 0 . . . * (17) Substitute from equation (26) into equation (17): AV n JJ + H 412G[ 1 1 +1 -1 q LA (27) k where G, J and H are defined by equations (23a), (26a), (26b) re.- 32 spectively, and are dependent only on c and D. In considering circular and square tulular columns, it was necessary to establish a range of -MM t limiting values of vausofd 1 it / 3 and values, it 1 d1t -g -202 1tb is now necessary to establish ranges for will be recalled that were adopted. tb Similarly, c and w D, the 1t constants defining the shape of the initial section, The range of 4 rn to b D n 20. b t unstiffened columrft cross D values considered here is from b D These limiting values were arrived at by making a survey of the angles available in the A.I.S.C. Manual of Steel ConO. struction. t 1 * y d L The range of b considered here is from t D IfT more or less arbitrary. to - 1 , the initial column cross sec. 20 and --- tion would be a closed square. - The other limiting value It can be argued that for buckling of the stiffening plate may occur. t is 1 <=-.- , local However, the stress in- tensity in the stiffening plate may be well below the proportional limit and it is hard to determine at just what value of buckling of plate becomes critical. t 1 t 40 1 lems. 1 T to -- t local In any case, the range of covers a considerable number of practical prob-- If J, H, and G for each of the four possible limiting conditions, l =--, t W bD4- -M 0M - Da 20i -=a namely 1 b D a 4; and T 1' b 40d4 t 1 -= d1 125 definitions (J, equation 1 b 1 1 b bn D a 20; are computed from their If k and q 1 125 (.26a), H, equation (26b), G, (23a)), and subox stituted into equation (27), four distinct expressions for a function of t will result. AV as These four equations and the 33 cross plots of k versas q could be used to find four optimum curves by trial, the method being identical with that used to find optimum curves for circular and square tubular columns. The com. putations involved in this approach are considerable, and a method which depends on the two optimum curves already determined for tubular columns and minimization criteria is used to show that the average optimum curve already determined for square and circular tubular columns can be used for the structural cross section under consideration. The derivation of the explicit form of minimization criteria for the cross section defined in Figure 4 follows: (27) with q Consider equation replaced by f(k), being careful to note that q n f(k) is the same function which applies to circular and square tube cross sections, that is, the cross plots of k versus q are independent of the shape of the cross section considered, and depend only on the graph representing a general solution of the characteristic e.- quation (13). (See Figure 5.) Then AV [J + H V12G[ lJ+l .-l) f (k) IA k r 1 dAV J S J f (k)+H 12G-- 1] + 1 ft(k) k f (k) H 1(12G[- ft(k) Jj+1)-1/2(24Gk--3 34 Let 0 and multiply both sides by 12G[ , k 1] + 1 Then 0 M f' (k)[(J-) 412G[k + k 12H H(12G[ 1 kk from equat ion (26a) J n1- H H. 1. a J Therefore, f(k) a g(k,c,D) .. +12G[ 77k 13+1 7 k [ f(k) - ft(k) 1+1 1 f'(k) . . . . . . . . . . . . (28) where g(kcD) 2G [" 412G[ . 1] + 1 + 12G[ 1 k 13 + 1] ..(28a) k and G is defined by equation (23a). If for k = 0.5, 0.7, 0.9, 1.00, g(kc,D) is computed using equation (28a) for each of the four possible combinations of the limiting vales of c M d1 and D n Table 1 and tabulated as follows. , the results can be added to (See Appendix H.) Table of Minimization Criteria G(kB) k B g(kcD) 1 1__ B=a TO- 2 _Dm4 100 Dm20 00 0 "T C D 4 0 Dm 20 0 0.90 0.08995 0.08573 0,08593 0.08650 0.08552 0.08569 0.70 0.20972 0,18022 0.18061 0.18642 0.17873 0.17960 0.50 0.24961 0.19267 0.19348 0.20815 0.18799 0.19092 Table 2. 35. It has already been shown (see Table 1) that for the f(k) fum tions involved, a difference of 25.75% in the minimization criteria does not change the location of the optimum stiffening curve on the graph representing a general solution of the characteristic equation appreciably. conclusion, A study of Table 2 leads to the following Since the minimization critera for the structural cross section defined in Figure 4 lie between or slightly below the minimization critexit for the two optimum curves obtained by trial compu- tations for tubular columns, it is concluded that the four optimum curves for the structural cross section in Figure 4 are very well (See approximated by the one average curve already determined. Figure 5.) A spot check of this conclusion is made for c -i D 4 , at the l]rel \f 1 Equation (27)1with J, H, a 0.750. or and G evaluated, reduces to AV- [.s557.o3 + 558.03VO.OO51469[ 1 k2 aV = 558.03 [ V.0051469 . ]+ ]. 1 - 1]+1 .1) q - 1] q L A1 LA1 . . (29) k If using the cross plot of k versus q for fI or = 0.750 and equation (29) the kq solution associated wi. th a minimum ution is found by trial, &V the minimum point thus found checks with that given by the one average curve drawn for tubular columns. Appendix F.) solo (See 36 G. Results and Discussion The graph of the general solution of the characteristic equation of the simple stiffened column system defined in Figure 1 and Part II, C, Scope, has been developed. An average optimum stiffening curve is superimposed on the general solution of the characteristic equation, For columns which conform to the limitations of Part II, C, Scope, and have cross sections which are within the ranges and of the types considered, the optimum stiffening solution may be readily computed using Figure 5. Although the location of the optimum curve depends on the shape of the cross section in a strict theoretical sense, the results of this investigation show that the location of the optimum curves for various cross sections does not vary appreciably. The introduction of an average optimum curve is reasonable, particuh rly when the flat character of the sidered. AV function near the minimum point is con The average curve presented here may well be applicable to stiffening problems involving far more complicated structural cross sections with two axes of symmetry. There are two classes of problems that can be solved using the optimum stiffening curve. The problems of the first class are those where an existing column is to be stiffened in order to be able to carry more load. The second class of problem occurs when a new col.. umn is being designed using optimum column stiffening ideas. Class lt As an example of problems of the first class, consider an ex- I1 37 isting circular tube column lot long with di - 3" and t = 1". What is the ultimate load that this column 30 x 106.) (Steel E can be made to support, and what is the optimum stiffening solution which will prevent buckling failure under the ultimate load? Compute A I (9 - 1) a 6.28 in 2 4(d12 00 d0 2) i k and I 1 (d 164 4 4.4 3.93 in. (81 - 1) ) d 064 1 Check radius of gyration, to make sure initial column is in Euler range, 0.791" r L-1 x 12 79 1 Assume that f YJP. a152 - fP P. L. > . 100 O.K. r1 = f respect to direct stress is = 30,000f/in 2 . a- # S.F. - Then the S.F. with 1.00. -W Therefore, f Por a S.F. P2 = 1.00 P2 T Compute P P.L. p * ~~or Por Pi and f 2 2 SEI L2 f 1 A1 1 y x 30 x 10 6 x 3.93 - 80,900-# 1202 _ 80, 00 - 12 , 800/in2 6*28 38 or P.L. 30,000 n 288 P~ M . 1 P 2.55 1 or - 188,733# Using Figure 6, find the optimum stiffening solution = 2.33 = q - 0.64 Optimum solution 0.654 or P1 1 Prove 1.528 P q = 0.64, k w 0.615 k = 0.615. , is a solution which increases the strength as required. = -L(l.q) S2 U = Por x 120 (1 "0.64) 188,733 30x10 6 x 3.93 x 120 x 0.36 x 0.040 = 0.864 2U -n t 12 2 x 0.864 j2 S 0. 36 x Tr 2.33 2.3 Compute volume of material added for optimum solution. k - 0.615 I2 k2 M l 2 T (d2 10.39 in0 n 64 0.378 - k - 0.615 2' d 40 ) T (d 4 64 2 d 2 = 3.82 in A 2 0. A, n 2 ?lr t = 2 1 M 1.705 x 0.41 2 A2 - A 1 a 4.40 in AV-= (A2 - A1 ) qL a 4.40 x 0.64 x 120 &V = 338 in 3 1 0. K. q a 0064, 39 Co mpute the volume of material added for constant cross section st iffeninge I2 2.33 G. 16 in 2.33 x 3.93 I 4 2 T (dd 4 64 2- d4) (d ~264 u2 0 1) d2 = 3.70 in. A " A 2 1 A2 . A = 2 Wr t = 2 W x 1.675 x 0.35 m = 3.68 AV = (A2 '- Al)L = 3.68 x 120 &V = 442 in3 Compute the percentage reduction of material added by using optimum stiffening instead of constant cross section. % reduction AV = 442 % reduction AV = 23.5 - 338 % 44E2 q = 0.64, k a 0.615 is the optimum solution, run through To show that the design of stiffened columns represented by points a little to either side of the optimum point. P7-- = 0.654 Pr (A) q a 0.59 k = 0.595 (B) q a 0.69 k = 0.630 Prove (A) and (B) are solutions which increase the strength as re.m quired. (A) q) L (17 x 120 (0,41)(0.040) = 0.985 1=T or . 2U 1 I (1q j2 [2 x 0.985j T 0.4J. 2 3 3 0.K. 40 (B) 1 U =-f x 120 x 0.31 (0.040) = 0.744 P or 2 P 0. 031) L 2 = 2.33 O.K. '-23 Compute the volume of material added for solution (A). k a 0.595 -3 9 k2 = 0.354 11.1 in4 d2 - 3.89 in A - A, W4.82 in2 V a 342 in3 > 338 in3 Compute the volume of material added for solution (B). k = 0.630 3.43 k2 n 0.397 4 9.90 in d2 = 3.78 in A 2 0 A, = 4.15 in 2 V = 344,in Therefore, > 338 in3 q a 0.64, k w 0.615 has been shown to be the optimum manner in which to stiffen the given column to its ultimate Po or 188#733#. Class 2f Problems of the second class are those where a new column is to be designed using optimum stiffening ideas. These problems can be re duced to problems of the first class by designing the initial cross 41 section first. In designing the initial section stress A, up to the fa, temporarily ignoring buckling failure. The shape (square tube, circular tube, or four angle section) will have to be decided upon from architectural considerations or arbitrarily. The most efficient initial cross section will be that with the highest practicable radius of gyration, which is the same as saying the smallest practicable t ratio. The smallest t that is to be permitted in any problem 1 1 must be fixed. Some of the factors influencing the engineerts judgr* ment in this matter are listed here. 1. The theoretical limit for L values, below which local buckling failure of the tube becomes critical. 2. The lower limiting -g value of available manufactured sizes. 3. The danger of damage to tubes with low t 1 values due to handling, erection, and fabrication is considerable. 4. Lateral clearance considerations. It is not desirable that any column take up too much room by having a large d small L so as to get a ratio. Then with the shape, Alland d all established, the initial cross section is completely defined, hence the problem remaining is of the first class. It is important to check whether the design of the initial cross section is in the Euler range before using the optimm stiffening curve. The reason for desiring the smallest t - practicable should be fully realized. The percentage saving in stiffening material is higher for less efficient heavy walled columns, but the total volume of material used is greater. Therefore, the design which shows the greatest overall economy of material can be 42 achieved by stiffening the most efficient initial cross section prac- ticable. U H. Conclusions From the results of this investigation, it may be concluded that: 1. A readily applicable method for determining the optimum stiffening solution for a column which is within the limitations of Part II, C, Scope, and has a cross section of the type, and within ranges, con.sidered, has been devised. 2. The main source of error in the computational work is in the k versus q cross plots. ditional This could be improved upon by computing ad- curves, and by using larger scale k plots, but this was not considered necessary. k versus q cross The large number of significant figures carried in all computations is an attempt to introduce no further error over and above that inherent in the versus 3. q k cross plots. Although savings of material are achieved by optimum stiffening design, the cost of fabrication probably outbalances any economy of material. In designs where weight is a more critical factor than in most civil engineering structures, optimum column stiffening might well find application. 4. The ideas successfully applied to the simplest type of column in this investigation may be applicable to a large number of more conm plicated column types. (A.) Optimum stiffening solutions for various other end conditions. (1) Fixed, symmetrical case. 43 (2) One end fixed, one pinned, unsymmetrical. (3) Combinations of end conditions involving partial fixity. (B) Optimum stiffening in the inelastic range of buckling failure. (C) Optimum stiffening for eccentrically loaded columns, and columns with initial curvature. 44 APPENDIX A Notation, L - length of the unstiffened column r = radius of gyration of the unstiffened colUimnts cross section =proportional limit stress intensity of the material used f M = resisting moment E n Modulus of Elasticity I = Moment of Inertia EI= bending stiffness P- radius of curvature P, Euler buckling resistance of unstiffened column Por= ultimate buckling resistance of stiffened column = the moment of inertia of the unstiffened cross section I the total moment of inertia of the stiffened cross section I? kmJ q - 2 the ratio of the stiffened length to the total length of the column L AV n the volume of the stiffening material A - the area of the unstiffened cross section A2 n the total area of the unstiffened cross section K = a constant depending only on the shape of the initial unm stiffened cross section fY - the yield point stress intensity of the material used S.F. = safety factor U L a, (1 -a q) c a convenient parameter 45 Notation (continued)t d = inside diameter of the initial tubular column d = outside diameter of the initial tubular column d2 a outside diameter of the stiffening sheet for tubular columns d 2 d B a ad lo= wall thickness of the initial tubular column " d d t a convenient constant depending on the 0 the unstiffened column b = length of angle leg C convenient notation - b D = T convenient notation ratio of 46 APPENDIX B Discussion of Dr. Dinnikts Paper A paper entitled Design of Columnr of Varying Cross Section, by Dr. A. Dinnik, appears in the 1932 Transactions of the A.S.M.E., in the Applied Mechanics Division. a stability coefficient In this paper, Dr. Dinnik defines for stiffened columns identical to those K dealt with in this investigationg EI . . . . . (2) Dr. Dinnikts paper or L2 This is the equation which Dr. Dinnik uses to define his stability coefficient K, in terms of the notation of this investigation. Equation (15) of this investigation can be written as followss r or 1 2E 2U oq i 2 2 2kU 1 ..i2q Therefore, according to this investigation, 2 . 2kU g-- q If we solve this expression for U, 0 ) ( U W * . . . . . . . . . . (30) Using K values from Table 4 of Dr. Dinnikts paper, and the proper associated values of equation (30). k and q a set of roots U can be obtained from Introduce this set of roots U, with the proper associ.. ated values of k and q, into the characteristic equation (13). The The com- characteristic equation (13) should reduce to an identity. putations here outlined follow in tabulated form. tan U tan kq T -q U = 1 k . . . . . . . . . . (13) (1) q - 0.2 12 n T (2) (3) I K (4) (5) U=(2)(3) tan (4) (7) (6) k ( (8) (9) tan (7) (5)x(S) (10) 1/km 0.01 0.15 15.0 3.88 0.4 1.55 48.078 0.1 0.039 0.0390 1.66 10 0.10 1.47 14.7 3.84 0.4 1.537 29.372 0.316 Q.121 0.1216 3w56 3.16 0.20 2.80 14.0 3. 74 0.4 1.498 14.100 0.450 0.169 0.1706 2.40 2.22 0.40 5.09 12.72 3.57 0.4 1.43 7.055 0.634 0.226 0.2300 1.62 1.58 0.60 6.98 11.63 3.42 0.4 1.37 4.913 0.775 0.265 0.2714 1.33 1.29 0.80 8.55 10.70 3.28 0.4 1.31 3.747 0.895 0.293 0.3017 1.13 1.12 (7) (6).(4) 1.5 q - 0.4 0.01 0.27 27.0 5.20 0.3 1.560 92.620 0.10 2.40 24.0 4.90 0.3 1.470 9.887 0.20 4.22 21.1 4.60 0.3 1.380 0.40 6.68 16.7 4.10 0.3 0.60 8.19 13.65 3.70 0.80 9.18 11.46 3.39 0.1 0.104 0.1044 9.63 10 0.316 0.309 0.3192 3.16 3.16 5.177 0.450 0.414 0.4394 2.28 2.22 1.230 2.820 0.634 0.520 0.5726 1.1615 1.58 0.3 1.110 2.014 0.775 0.573 0.6452 1.30 1.29 0.3 1.017 1.628 0.895 0.608 0.6959 1.13 1.12 I (1) K q w 0.6 (2) (3) /"S (4) (5) U=(2).(3) tan (4) 0.01 0.60 60.0 7.75 0.2 1.55 48.078 0.10 4.50 45.0 6.70 0.2 1.34 0.20 6.69 33.45 5.79 0.2 0.40 8.51 21.28 4.62 0.60 9.24 15.40 0.80 9.63 12.04 (6) (7) 1.5()4) (8) tan (7) (9) (5)x(8) (10) 1/k 0.1 0.232 0,2363 11.35 10 4.256 0,316 0.635 0,7368 3.14 3.16 1.158 2.283 0.450 0.780 0.9893 0.2 0.924 1.324 0.634 0.879 1.2072 1.60 1.58 3.93 0.2 0.786 1.001 0.775 0.915 1.2997 1,30 1.29 3.48 0.2 0.696 00835 0.895 0.935 1,3550 1.13 1.12 2.26 2,22 (7) q - 0.8 4-(6).(4) 0.01 2.26 226.0 0.10 8.59 0.20 15.05 0.1 1.505 15.1765 85.9 9.27 0.1 0.927 9.33 46.65 6.85 0.1 0.40 9.67 24.18 4.92 0.60 9.78 16.30 0.80 9.84 12.30 0.1 0.602 0.6871 1.3525 0.316 1.171 2o3666 3.16 3.16 0.685 0.8170 0.450 1.231 2.8288 2.31 2,22 0.1 0.492 0,5360 0.634 1.248 2.9896 1.60 1.58 4.05 0.1 0.405 0.4287 0.775 1.258 3.0920 1.32 1.29 3.51 0.1 0.351 0.3662 0.895 1.258 3,0920 1.13 1.12 10.4 10 0, 49 The results of checking the stability coefficients K presented in Table 4 of Dr. Dinnikts paper lead to the following discussion. The values of (9) and (10) in the tables of computations just prev* sented should be equal. For any discrepancy which may be conskbred too great, the value of U method, and the K may be improved by Newtonts iteration value accordingly modified, It would appear from the trend of the errors indicated by the check made here, that graphical methods were used to solve the characteristic equation while in transcendental form, and that refinement by iteration was not carried out, in Dinnik's paper. Values of U taken from Table 15 of Dr. Dinnikts paper, when sub. stituted into K do not give the . . . . . I K Dr. Dinnikts values of Dr. Dinnik's Table 4. (47) However, the K values of Table 4 in Dr. Dinnikts paper have been checked and shown to be approximately correct. Furthermore, values of U from Dr. Dinnikts Table 15 do not satisfy his equation (46). inasmuch as the values of K Therefore, tabulated in Table 4 of Dr. Dinnik t s paper check when substituted into the derivation presented in this investigation, it is suggested that equations (46) and (47), and Table 15 of Dr. Dinnikts paper should be revised as follows. Note that the suggested corrections are presented in terms of Dr. Dinnikts notation. The only difference in notation beingt 50 Dinnik This Investigation q L 7% I i I2 L The sign conventions are identical. Equation (46) should be re. vised to read tan-A tan U U 1 Equation (47) should be revised to read 2kU p or (1. -2 j)A) T EI 2 t Table 15 should be revised to reads -0.z U.4 U.6 068 0.01 1.55 1.56 1.55 1.51 0.1 1.54 1.47 1.34 0.93 0.2 1.50 1.38 1.16 0.69 0.4 1.43 1.23 0.92 0.49 0.6 1.37 1.11 0.79 0.41 0.8 1.31 1.02 0.70 0.35 Table of U Values 51 APPENDIX C (1) Computations fbr the plot which represents the general solu tio n of the characteristic equation (13). tan U tan 1 -q lr(1 - U I k . . . . . . 0 0 0 0 q) 0 cr2U cr 0 0 0. . . . . . . . . (13) ** ** (16) "V 11 k - 0,950 (1) U tan U (2) 6 (3) (4) (5) kU tali- (2) (4)/(3) (6) (7) q (8) 1 - (6) (9) (7)/(8)4 or 0.0000 0.0000 0.2000 0.2027 5.1928 0.1900 0.4000 0.4228 2.4895 0.6000 0.6841 0.7500 0.0000 W/2 00 1.0000 0.0000 0.0000 0.9500 1.38055 7.2660 0.8790 0.1210 0.1273 0.9605 0.3800 1.1888 3.1284 0.7578 0.2422 0.2546 0.9513 1.5386 0.5700 0.9945 1.7447 0.6357 0.3643 0.3820 0.9537 0.9361 1.1244 0.7125 0.8439 1.1844 0.5422 0.4578 0.4775 0.9587 0.8500 1.1393 0.9239 0.8075 0.7459 0.9237 0.4802 0.5198 0.5411 0.9606 0.9500 1.3984 0*7527 0.9025 0.6452 0.7149 0.4169 0.5831 0.6048 0.9641 1.0500 1.7433 0.6038 0.9975 0.5432 0.5446 0.3526 0.6474 0.6685 0.9684 1.1500 2.2345 0.4711 1.0925 0.4030 0.2872 0.7128 0.7321 0.9736 1.2500 3.0096 0.3497 1.1875 0.3364 0.2833 0.2208 0.7792 0.7958 0.9791 1.3500 4.4552 0.2363 1.2825 0.2320 041809 0.1532 0.8468 098594 0.9853 1.4000 5.7979 0.1815 1.3300 0.1795 0.1350 0.1189 0.8811 0.8913 0.9886 1.4500 8.2381 0.1278 1.3775 0.1271 0.0923 0.0845 0,9155 0,9231 0.9918 0.4403 1.5000 14.101 0.0746 1.4250 0.0745 0.0523 0.0497 0.9503 0.9549 019952 1.5500 48.078 0.0219 1.4725 0.0219 0.0149 0.0147 0.9853 0.9868 0.9985 1.5600 92.620 0.0114 1.4820 0.0114 0.0077 0.0076 0.9924 0.9931 0.9993 00 0.0000 1.4923 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 W/ 2 C') I k * 0.90 U tan U (2) ta111 tan U (4) (3) W tali (2) 0.0000 0.1000 0.0000 0.1003 11.0777 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.2027 0.3093 0.4228 0.5463 0.6841 0.8423 5.4814 3.5923 2.6279 2.0339 1.6241 1.3101 0.1800 0.2700 0.3600 0.4500 0.5400 0.6300 1.3904 1.2993 1.2072 1.1138 1.0189 0,9221 0.7500 0.9316 1.1926 0.8000 0.8500 0.9000 0.9500 1.0000 1.0500 1.1000 1.1500 1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 1.5000 1.5500 1.0296 1.1393 1.2602 1.3984 1.5574 1.7433 1.9648 2.2345 2.5722 3.0096 3.6021 4.4552 5.7979 8.2381 14.101 48.078 1*0792 0.9752 0.8817 0.7946 0.7134 0.6373 0.5655 0.4972 0.4320 0.3692 0.3085 0.2494 0.1916 0.1349 0.0788 0.02311 1.5600 92.620 00 @/2 0.0000 I/2 0.0900 1.4808 (5) (4)/(3) (6) (5) 2 1+(s) (7) q (8) 1 - (6) (9) (7 F/ii (8)(8) vpr or 1.0000 0.9427 0,0000 0.0573 0.0000 0.06366 0.9000 0.9001 7.7244 4.8122 3.3533 2.4751 1.8868 1.4637 0.8854 0.8279 0.7703 0.7122 0.6536 0.5941 0.1146 0.1721 0.2297 0.2878 0.3464 0.4059 0.1273 0.1910 0.2546 0.3183 0.3820 0.4456 0.9003 0.9011 0.9022 0,9042 0.9068 0.9109 0.6750 0.8730 1.2933 '0.5639 0.4361 0.4775 0.9133 0.7200 0.7650 0.8100 0.8550 0.9000 0.9450 0.9900 1.0350 1.0800 1.1250 1.1700 1.2150 1.2600 1.3050 1.3500 1.3950 0.8235 0.7728 0.7226 0.6714 .6197 0.5674 0.5147 0.4614 0.4078 093537 0.2992 0.2445 0.1893 0.1341 0.0787 0.02311 1.1438 1.0102 0.8921 0.7855 0.6866 0.6004 0.5199 0.4458 0.3776 0.3144 0.2557 0.2012 0.1502 0.1028 0.0583 0.0166 0.5335 0.5025 0.4715 0.4399 0.4071 0.3752 0.3421 0.3083 0.2741 0.2392 0.2036 0.1675 0.1306 0.0932 0.0551 0.0163 0.4665 0.4975 0.5285 0.5601 0.5929 0.6248 0.6579 0.6917 0.7259 0.7608 0.7964 0.8325 0.8694 0.9068 0.9449 0.9837 0.5093 0.5411 0.5730 0.6048 0.6366 0.6685 0.7002 0.7321 0.7639 0.7958 0.8276 0.8594 0.8913 0,9231 0.9549 0.9868 0.9160 0.9195 0.9223 0.9261 0.9314 0.934:7 0.9396 0.9448 0.9503 0.9560 0.9623 0.9687 0.9754 0,9823 0.9895 0.9968 0.01200 1.4040 0.01200 0.0085 0.0084 0.9916 0.9931 0.9985 0.0000 0.0000 1.0000 1.0000 1.0000 0.0000 1.4137 0.0000 00 16.453 ..... ..... .- I..., . .1 -. 1 1 111.." .." .1. I i 11 . I . . (1) k = 0.850 U 11 1 ii6b (2) tan U 1.1765 0..0000 0.0000 0.2000 0.2027 0.4000 (3) (4) I tann()((() iT/ 5.8041 0.1700 0.4228 2.7826 0.6000 0.6841 0.7500 00 (2) (5) (4)/(3) (6) (5) (7) q 1(6) (8) (9) 2U( i 0*00o0 (8) or 00 1.0000 0.0000 0.0000 0.8500 1.4002 8*2364 0,8917 0.1083 0.1273 0.8507 0.3400 1.2258 3.6052 0.7829 0.2171 0.2546 0.8527 1.7197 0.5100 1.0441 2.0472 0.6718 0.3282 0.3820 0.8592 0.9361 1.2568 0.6375 0.8987 1.4097 0.5850 0.4150 0.4775 0.8691 0.8500 1.1393 1.0327 0.7225 0.8015 1.1093 0.5259 0.4741 0.5411 0.8762 0.9500 1.3984 0.8413 0.8075 0.6994 0.8661 0.4641 0.5359 0.6048 0.8861 1.0500 1.7433 0.6749 0.8925 0.5937 0.6652 0.3995 0.6005 0.6685 0.8983 1.1500 2.2345 0.5265 0.9775 0.4846 0.4958 0.3315 0.6685 0.7321 0.9131 1.2500 3.0096 0.3909 1.0625 0.3726 0.3507 0.2596 0.7404 0.7958 0.9304 1.3500 4.4552 0.2641 1.1475 0.2582 0.2250 0.1837 0.8163 0.8594 0.9498 1.4000 5.7979 0.2029 1.1900 0.2002 0.1682 0.1440 0.8560 0.8913 0.9604 1.4500 8.2381 0.1428 1.2325 0.1418 0.1151 0.1032 0.8968 0.9231 0.9715 2 1.5000 14.101 0.0834 1.2750 0.,0832 0.0653 0.0613 0.9387 0.9549 0.9830 1.5500 48.078 0.0245 1.3175 0.0245 0.0186 0.0183 0.9817 0.9868 0.9948 1.5600 92.620 0.0127 1.3260 0.0127 0.0096 0.0095 0.9905 0.9931 0.9974 0.0000 1.3352 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 2 -- - (1) - - . . . I . - - .- Mmmm Ai -li k = 0.8 tan U 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 (2) tan (3) 500 U k U 0.0000 0.1003 0.2027 0.3093 0.4228 0.5463 00 12.4626 6.1667 4.0413 2*9564 2.2881 0.0000 0.0800 0.1600 0.2400 0.3200 0.4000 0.6000 9.6841 1.8272 0.7000 0.7500 0.8423 0.9316 1,4840 1.3417 0.8000 1.0296 1.2141 0.8500 0.9000 0.9500 1.0000 1.0500 1.1393 1.2602 1.3984 1.5574 1.7433 1.0972 0.9919 0.8939 0.8026 0.7170 1.1000 1.9648 1.1500 1.2000 1.2500 1.3000 1.3500 1.4000 2.2345 2.5722 3.0096 3.6021 4.4552 5.7979 8.2381 1.4500 1.5000 1.5500 1.5600 '/2 14.101 48.078 92.620 00 (4) tan (2) (5) (4)/(3) (6) 1+(5) (7) - q 1 - IT/2 1.4907 1.4100 1.3282 1.2446 1.1588 00 18.633 8..8125 5.5341 3.8893 2.8970 0.4800 1.0700 0.5600 0.6000 0.9778 0.9303 0.6400 0.6800 0.7200 0.7600 0.8000 0.8400 0.6362 0.8800 0.5666 0.5594 0.4860 0.4153 0.3470 0.2806 0.2156 0.9200 0.9600 1.0000 1.0400 1.0800 1.1200 0.5101 0.4524 0.3937 0.3340 0.2736 0.2124 0.1517 1.1600 0.1506 0.1298 0.1149 0.0886 0.0260 0.0135 0.0000 1.2000 .1,2400 1.2480 1.2566 0.,0884 0.0260 0.0135 0.0000 0.0737 0.0210 0.0108 0,0000 0.0686 0.0206 0.0107 0.0000 (6) (8) (9) - (1) (8) 1.0000 0.94906 0.8981 0.8470 0.7955 0.7434 0.0000 0.05094 0.1019 0.1530 0.2045 0.2.566 0.0000 0.06366 0.1273 0.1910 0.2546 0.3183 0.8000 0.8002 0.8005 0.8010 0.8032 0.8061 2.2229 0.6897 0.3103 0.3820 0.8123 1.7461 1.5505 0.6358 0.6079 0.3642 0.3921 0.4456 0.4775 0.8173 0.8212 0.8818 1.3778 0.5794 0.4206 0.5093 0.8258 0.8317 0.7814 0.7294 0.6763 0.6221 1.2231 1.0853 0.9597 0.8454 0.7406 0.5502 0.5205 0.4897 0.4590 0.4255 0.4498 0.4795 0.5103 0.5410 0.5745 0.5411 0.5730 0.6048 0.6366 0.6685 0.8313 0.8368 0.8437 0.8498 0.8594 0.6439 0.3917 0.6083 0.7002 0.8687 0.5545 0.4713 0.3937 0.3212 0.2533 0.1896 0.3567 0.3203 0.2826 0.2431 0.2021 0.1594 0.6433 0,6797 0.7175 0.7569 0.7979 0.8406 0.7321 0.7630 0.7958 0.8276 0.8594 0.8913 0.8787 0.8898 0.9016 0.9146 0.9284 0.9431 0.8851 0.9231 0.9589 0.9314 0.9794 0.9893 1.0000 0.9549 0.9868 0.9931 1.0000 0.9754 0.9925 0.9962 1.0000 orF 01 01 k w 0.750 (1) u tan U (2) 1.3333 tan U (3) (4) k U tan 1(2) 0.0000 0.0000 0.2000 0.2027 6.5777 0.1500 1.4199 0.4000 0.4228 3.1535 0.3000 0.6000 0.6841 1.9489 0.7500 0.9361 0.8500 0 (5) (4)/(3) 000ooo (6) 1+(5) (7) q 1-() (8) 2 U T (9) (7_ ()r 100000 0.0000 0.0000 0.7500 9s4660 0.9045 0.0955 0.1273 0,7502 1.2637 4.2123 0.8081 0.1919 0s2546 0.7537 0.4500 1.0967 2.4371 0.7091 0.2909 0.3820 0,7615 1.4243 0.5625 0.9587 1.7044 0.6302 0.3698 0.4775 0.7745 1.1393 1.1703 0.6375 0.8637 1.3548 0.5753 0.4247 0.5411 0.7849 0.9500 1.3984 0.9534 0.7125 0.7615 1.0688 0.5166 0.4834 0.6048 0.7993 1.0500 1.7433 0.7648 047875 0.6529 0.8291 0.4533 0.5467 0.6685 0.8178 1.1500 2.2345 0.5967 0.8625 0.5380 0.6238 0.3842 0.6158 0.7321 0.8411 1.2500 3.0096 0.4430 0.9375 0.4170 0.4448 0.3079 0.6921 0.7958 0.8697 1.3500 4.4552 0.2993 1.0125 0.2908 0.2872 0.2231 0.7769 0.8594 0.9040 1.4000 5.7979 0.2300 1.0500 0.2261 0.2153 0.1772 0.8228 0..8913 0.9231 1.4500 8.2381 0.1618 1.0875 0.1594 0.1466 0.1279 0.8721 0.9231 0.9448 1.5000 14.101 0.0946 1.1250 0.0943 0.0838 0.0773 0.9227 0.9549 0.9663 1.5500 48.078 0.0277 1.1625 0.0277 0.0238 0.0232 0.9768 0.9868 0.9899 1.5600 92.620 0.0144 1.1700 0.0144 0.0123 0.0122 0*9878 019931 0.9947 00 0.0000 1.1781 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 c0n cm k - 0.7 (1) U tan U (2) 14286 (3) (4) k U ta~ (5) (4)/(3) (2) ta7 (6) q (7) (8) 1.(6) 2U U1+(5) (9) M (8) m f or 0.0000 0.0000 0 0.0000 / 1.0000 0.0000 0.0000 0.1000 0.1003 14.2433 0.0700 1.5007 214385 0.95543 0.04457 0.06366 0.2000 0.3000 0.4000 0.5000 0.2027 0.3093 0.4228 0.5463 7.0479 4.6188 3.3789 2.6150 0.1400 0.2100 0.2800 0.3500 1.4299 1.3580 1.2831 1.2055 10.2135 6.4666 4.5825 3.4442 0.9108 048661 0.8209 0.7750 0.0892 0.1339 0.1791 0.2250 0.1273 0.1910 0.2546 0,3183 0.7007 0.7011 0.7034 .0.7069 0.6000 0.7000 0..6841 0.8423 2.0883 1.6961 0.4200 0.4900 1.1242 1.0381 2*6766 2.1185 0.7280 0.6793 0.2720 0.3207 0.3820 0.4456 0.7120 0.7197 0.7500 0.8000 0.8500 09000 0.9500 1.0000 1.0500 1.100 1.1500 1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 1.5000 1.5500 1.5600 27/2 0.9316 1.0296 1.1393 1.2602 1.3984 1.5574 1.7433 1.9648 2.2345 2.5722 3.0096 3.6021 4,4552 5.7979 8.2381 14.101 48,078 92.620 00 1.5335 1.3875 1.2539 1.1336 1.0216 0..9173 0.8195 0.7271 0.6393 0.5554 0.4747 0.3966 0.3207 0.2464 0.1734 0.1013 0.0297 0.0154 0.0000 0.5250 0.5600 0.5950 0.6300 0.6650 0.7000 0.7350 0.7700 0,8050 0.8400 0.8750 0.9100 0.9450 0.9800 1.0150 1.0500 1.0850 1.0920 1.0995 0.9930 0.9463 0.8976 0.8479 0.7961 0.7423 0.6855 0.628'? 0..5688 0.5070 0.4432 0.3776 0.3103 0.2416 0.1717 0.1010 0.0297 0.0154 0.0000 1.8914 1.6898 1.5085 1.3458 1.1971 1.0604 0.9326 0,8164 0.7065 0.6035 0.5065 0.4149 0.3283 0.2465 0.1692 0.0962 0.0274 0.0141 0.0000 0.6541 0.6282 0.6014 0.5737 0,5449 0.5147 0.4826 0.4495 0.4140 0.3764 0.3362 0.2932 0.2472 0.1978 0.1447 0.0878 0.0267 0.0139 0.0000 0.3459 0.3718 0.3986 0.4263 0.4551 0.4853 0.5174 0.5505 0.5860 0.6236 0.6638 0.7068 0,7528 0.8022 0.8553 0.9122 0.9733 0.9861 1.0000 0.4775 0.5093 0,5411 0,5730 0.6048 0.6366 0.6685 0.7002 0,7321 07639 0,7958 098276 0.8594 0.8913 0,9231 0.9549 0.9868 0.9931 1.0000 0.7244 0.7300 0.7366 0.7440 0,7525 0*7623 0.7740 0,7862 0.8004 0.8163 0.8341 0.8540 0.8760 0.9000 0.9265 0.9553 0.9863 0.9930 1.0000 00 2 0.7000 0.70013 Ul . ........ ... . k a 0.650 U8tan U (3) (2) (1) 8 1 00 k U 0.0000 (6) (4(5) tano (2) 1/2 (4)/(3) (7) (9) (8) 5 1..(6) 00 1.0000 0.0000 0.0000 0.6500 c (8) 0.0000 0.0000 0.2000 0.2027 7.5900 0.1300 1.4398 11.0753 0.9172 0.0828 0,1273 0.6504 0.4000 0.4228 3.6388 0.2600 1.3026 5.0100 0.8336 0.1664 0.2546 0.6536 0.6000 0.6841 2.2489 0.3900 1.1524 2.9548 0.7471 0.2529 0.3920 0.6620 0.7500 0.9361 1.6435 0.4875 1.0242 2.1009 0.6775 0.3225 0.4775 0,6753 0.8500 1.1393 1.3504 0.5525 0.9334 1.6894 0.6282 0.3718 0.5411 0*6871 0.9500 1.3984 1.1002 0.6175 0.8331 1.3491 0.5743 0.4257 0.6048 0.7039 1.0500 1.7433 0.8825 0.6825 0.7231 1.0595 0.5144 0.4856 0.6685 0.7264 1.1500 2.2345 0.6885 0.7475 0.6030 0.8067 0.4465 0.5535 0.7321 0.7560 1.2500 3.0096 0.5112 0.8125 0.4726 0.5817 0.3678 0.6322 0.7958 0.7944 1.3500 4.4552 0.3453 0.8775 0.3325 0.3789 0.2748 0.7252 0.8594 0.8438 14000 5.7979 0.2654 0.9100 0.2594 0.2851 0.2219 0.7781 0.8913 0.8730 1.4500 8.2381 0.1868 0.9425 0.1847 0.1960 0.1639 0.8361 0.9231 0.9058 1.5000 14.101 0.1091 0.9750 0.1087 0.1115 0.1003 0.8997 0.9549 0.9422 1,5500 48.078 0.0320 1.0075 0.0320 0.0318 0.0308 0.9692 0.9868 0.9822 1.5600 92.620 0.0166 1.0140 0.0166 0.0164 0.0161 0,9839 0.9931 0.9971 TT/ 2 00 0.0000 1.0210 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 Ai N k - 0.6 (1) U tan U (2) tan (3) (4) k U tan 1(2) (5) (4)/(3) (6) -5 q (7) (8) 1-(6) 2T (9) (8) 1+(5) 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000 1.0500 1.1000 1.1500 1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 o.oo 0.1003 0.2027 0.3093 0.4228 0.5463 0.6841 0.8423 0.9316 1.0296 1.1393 1.2602 1.3984 1.5574 1.7433 1.9648 2*2345 2.5722 3.0096 3.6021 4.4552 5.7979 -8.2381 0 25.178 12.082 7.707 5.510 4.180 3,281 2.626 2.3580 2.1196 1.9043 1.7100 1.5311 1.3655 1.2110 1.0659 0.9288 0.7986 0.6744 0.5556 0.4420 0.3333 0.2294 _./ 16.6168 8.2223 5.3885 3.9420 3.0508 2.4363 1.9787 1.7890 1.6188 1.4629 1.3225 1.1918 1.0702 0,9560 0.8483 0.7459 0.6480 0.5538 0.4627 0.3741 0.2875 0.2023 0.0600 0.1200 0.1800 0.-2400 0.3000 0.3600 0.4200 0.4500 0.4800 0.5100 0.5400 0.5700 0.6000 0.6300 0.6600 0.6900 0.7200 0.7500 0.7800 0.8100 0.8400 0.8700 1.5107 1.4498 1.3873 1.3224 1.2541 1.1813 1.1029 1.0611 1.0174 0.9712 0.9234 0.8727 0.8193 0.7629 0.7035 0.6409 0.5750 0.5058 0.4334 0.3580 0.2800 0.1996 1.0000 0.96179 0.9235 0.8851 0.8464 0.8069 0.7664 0.7242 0.7022 0.6794 0.6557 0.6310 0.6049 0.5773 0.5477 0.5159 0.4815 0.4440 0.4028 0.3572 0.3065 0.2500 0.1866 Pr c 0.0000 0.03821 0.0765 0.1149 0.1536 0.1931 0.2336 0.2758 0.2978 0.3206 0.3443 0.3690 0.3951 0.4227 0.4523 0.4841 0.5185 0.5560 0.5972 0.6428 0.6935 0,7500 0.8134 0.0000 0.06366 0.1273 0.1910 0.2546 0.3183 0.3820 0.4456 0.4775 0.5093 0.5411 0.5730 0.6048 0.6366 0.6685 0.7002 0.7321 0.7639 0.7958 0.8276 0.8594 0.8913 0.9231 0.6000 0.60022 0.6009 0.6016 0.6033 0,6067 0.6115 0.6189 0.6237 0.6295 0.6363 0.6440 0.6533 0.6640 0.6766 0.6914 0.7082 0.7279 0.7504 0.7767 0.8070 0.8415 0.8811 1.5000 14.101 0.1182 0.9000 0.1177 0.1308 0.1157 0.8843 0.9549 0,9261 1.5500 48.078 0.0347 0.9300 0.0347 0.0373 0.0272 0.9728 1.5600 0.9868 92.620 0.9858 0.0180 0.9360 0.0180 0.0192 0.0188 0.9812 0,9931 0.9880 "/2 00 0.0000 0.9425 0.0000 0.00001 0.0000 1.0000 1.0000 1.0000 cmi k - 0.575 (1) (2) (3) ,7391 kU (4) (5) tan~ (2) (4)/(3) (6) -q (7) (8) (9) U tan U 0.0000 0.0000 00 0.0000 '/2 00 1.0000 0.0000 0.0000 0.5750 0.2000 0.2027 8.5796 0.1150 1.4548 12.6504 0.9267 0.0733 0.1273 0.5758 0.4000 0.4228 4.1132 0.2300 1.3323 5.7926 0.8528 0.1472 0.2546 0.5782 0.6000 0.6841 2.5421 0.3450 1.1960 3.4666 0.7761 0.2239 0.3820 0.5861 0.7500 0.9361 1.8578 0.4313 1.0770 2.4971 0.7140 0.2860 0.4775 0.5990 0.8500 1.1393 1.5265 0.4888 0.9909 2.0272 0.6697 0.3303 0.5411 0.6104 0.9500 1.3984 1.2436 0.5463 0.8936 1.6357 0.6206 0.3794 0.6048 0.6273 1.0500 1.7433 0.9976 0.6038 0.7842 1.2988 0.5650 0.4350 0.6685 0.6507 1.1500 2.2345 0.7783 0.6613 0.6614 1.0002 0.5000 0.5000 0.7321 0.6830 1.2500 3.0096 0.5779 0.7188 0.5240 0.7290 0.4216 0.5784 0.7958 0.7268 1.3500 4.4552 0..3904 0.7763 0.3722 0.4795 0.3241 0.6759 0.8594 0.7865 1.4000 5.7979 0.3000 0.8050 0.2915 0.3621 0.2658 0.7342 0.8913 0.8237 1.4500 8.2381 0.2111 0.8338 0.2081 0.2496 0.1997 0.8003 0.9231 0.8670 1-(6) (8) 1.5000 14,.101 0.1233 0.8625 0.1227 0.1423 0.1246 0. 8754 0.9549 0.9167 1.5500 48.078 0.0362 0.8913 0.0362 0.0406 0.0390 0.9610 0.9868 0.9739 1.5600 92.620 0.0188 0*8970 0.0188 0.0210 0.0206 0.9794. 0.9931 0.9862 00 0.0000 0.9032 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 2 0 k - 0.55 (1) (2) (3) tan U tan-18U k U (4) tan' (2) (5) (4)/(3) (7) (6) () 1*q5 q (9) (8) 1..(6) (8) 0.0000 0.0000 00 0.0000 1/2 00 1.0000 0.0000 0.0000 0.5500 0.2000 0.2027 8.9699 0.,1100 1.4598 13.2709 0.9299 0.0701 0.1273 0.5507 0.4000 0.4228 4.3003 0.2200 1.3423 6.1013 0.8592 0.1408 0.2546 0.5530 0.6000 0,6841 2.6577 0.3300 1.2109 3.6693 0.7858 0.2142 0.3820 0.5607 0.7500 0.9361 1.9423 0.4125 1.0953 2.6552 0.7264 0.2736 0.4775 0.5730 0.8500 1.1393 1.5959 0.4675 1.0111 2.1627 0.6838 063162 0.5411 0.5844 0.9500 1.3984 1.3002 0.5225 0.9152 1.7516 0.6366 0.3634 0.6048 0.6009 1.0500 1.7433 1.0430 0.5775 0.8064 1.3964 0.5827 09.4137 0.6685 0.6188 1.1500 2.2345 0.8137 0.6325 0.6830 1.0798 0.5192 0.4808 0.7321 0.6567 1.2500 3.0096 0.6041 0.6875 0.5434 0.7904 0.4415 0.5585 0.7958 0.7018 1, 3500 4.4552 0.4103 0.7425 0.3894 0.5244 0.3440 0.6560 0.8594 0.7633 1.4000 5.7979 0.3136 0.7700 0.3039 0.3947 0.2830 0.7170 0.8913 0.8044 1.4500 8.2381 0.2207 0.7975 0.2172 0,2724 0.2141 0.7859 0.9231 0.8514 1.5000 14,101 0.1289 0.8250 0.1282 0.1554 0.1345 0..8655 099549 0.9064 1.5500 48.078 0.0378 0.8525 0.0378 0.0443 0.0424 0.9576 0.9868 0.9704 1.5600 92.620 0.0196 0.8580 0.0196 0.0228 0.0223 0.9777 0.9931 0.9845 0* 0.0000 0.8639 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 2 -P m) I-j k = 0.525 (1) U tanU (2) 1,9048 tn (3) k U (4) an~A(2) (5) (6) (5) (4)/(3) 1+(5) (7) 1-(6) (9) (8) C?() 1 r ( ()o 00 1.0000 0.0000 0.0000 0.5250 1.4648 13.9504 0.9331 0.0669 0.1273 0.5255 0.2100 1.3524 6.4400 0.8656 0.1344 0.2546 0.5279 2.7843 0.3150 1.2260 3.8920 0.7956 0.2044 0.3820 0.5351 0.9361 2.0348 0.3938 1.1140 2.8288 0.7388 0.2612 0.4775 0.5470 0.8500 1.1393 1.6719 0.4463 1.0318 2.3118 0.6980 0.3020 0.5411 0.5581 0.9500 1.3984 1.3621 0.4988 0.9375 1.8795 0.6527 0.3473 0.6048 0.5742 1.0500 1.7433 1.0926 0.5513 0.8296 1.5048 0,6008 0.3992 0.6685 0.5972 1.1500 2.2345 0.8525 0.603q 0.7059 1.1691 0.5390 0.4610 0.7321 0.6297 1.2500 3.0096 0.6329 0.6563 0.5643 0.8598 0.4623 0.5377 0.7958 0.6757 1.3500 4.4552 0.4275 0.7088 0.4040 0.5700 0.3631 0.6369 0.8594 0.7411 1.4000 5.7979 0.3285 0.7350 0.3174 0.4318 0.3016 0.6984 0.8913 0.7836 1.4500 8.2381 0.2312 0.7613 0.2272 0.2984 0.2298 0.7702 0.9231 0.8344 0.0000 0.0000 00 0.0000 0.2000 0.2027 9.397li 0.1050 0.4000 0.4228 4.5052 0.6000 0,6841 0.7500 U/2 1.5000 14.101 0.1351 0.7875 0.1343 0.1705 0.1457 0.8543 0,9549 0,8946 1.5500 48,078 0.0396 0.8138 0.0396 0.0487 0.0464 0.9536 0.9768 0.9664 1.5600 92.620 0.0206 0.8190 0.0206 0.0252 0.0246 0.9854 0.9931 0.9822 00 0.0000 0.8247 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 2 or k n 0.5 (1) U (2) tan U (3) (4) k U (5) tan~ (2) (6) (4)/(3) tnU1+(5) (7) q 1.2U (8) .( T.(6 (9) (7)i (8) or 0.0000 .0000 00 0.0000 "/2 00 1.0000 0.0000 0.0000 0.5000 0.1000 0.1003 19.9401 0.0500 1.5207 30.414 0.9681 0.0318 0.06366 0.,50003 0.2002 0.3000 0.4000 0.5000 0.6000 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000 1.0500 1.1000 1.1500 1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 1.5000 1.5500 1.5600 W/9 0.2027 0.3093 0.4228 0.5463 0.6841 0.8423 0.9316 1.0296 1.1393 1.2602 1.3984 1.5574 1.7433 L.9648 2.2345 2.5722 3.0096 3.6021 4.4552 5.7979 8.2381 14.101 48.078 92.620 9.8668 6.4662 4.7304 3.6610 2.9235 2.3745 2.1468 1.9425 1.7555 1.5870 1.4302 1.2842 1.1472 1.0179 0.8951 0.7775 0.6645 0.5552 0.4489 0.3450 0.2428 0.1418 0..0416 0.0216 0.0000 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.3750 0.4000 0.4250 0.4500 0.4750 0.5000 0.5250 0.5500 0.5750 0.6000 0.6250 0.6500 0,6750 0.7000 0.7250 0.7500 0.7750 0.7800 1.4699 1.4174 1.3625 1.3042 1.2412 1.1722 1.1349 1.0954 1.0530 1.0085 0.9606 0.9092 0.8530 0.7943 0.7301 0.6609 0.5865 0.5068 0.4219 0.3322 0.2382 0.1409 0.0416 0.0216 0.0000 14.699 9.449 6.813 5.217 4.137 3.349 3.026 2.739 2.478 2.241 2.022 1.818 1.6265 1.4442 1.2697 1.1015 0.9384 0.7797 0.6250 0.4746 0.3286 0.1879 0.05367 0.02769 0.0000 0.9363 0.9043 0.8720 0.8392 0.8053 0.7701 0.7516 0.7325 0.7125 0.6915 0.6691 0.6451 0.6193 0.5909 0.5594 0.5241 0.4841 0.4381 0.3846 0.3218 0.2473 0.1582 0.05094 0.02694 0.0000 0.0637 0.0957 0.1280 0.1608 0.1947 0.2299 0.2484 0.2675 0.2875 0.3085 0.3309 0.3549 0.3807 0.4091 0.4406 0.4759 0.5159 0.5619 0.6154 0.6782 0.7527 0.8418 0,9491 0.9731 1.0000 0.1273 0.1910 0.2546 0.3183 0.3820 0.4456 Q.4775 0.5093 0.5411 0.5730 0.6048 0.6366 0.6685 0.7002 0.7321 0.7639 0.7958 0.8276 0.8594 0.8913 0.9231 0.9549 0.9868 0.9931 1.0000 0.5004 0.5010 0.5028 0.5052 0.5097 0.5159 0.5202 0.5252 0.5313 0.5384 0.5471 0.5575 0.5695 0.5842 0.6018 0.6230 0.6483 0.6789 0.7161 0.7609 0.8154 0,8814 0.9618 0.9798 1.0000 1r/ 4 WAN IIIi I II, IAL -. j J Ik,.,, APPENDIX C, (a) I.500 G4 FlGURE *5~ 01 520 SQ 1~ t. Q:7S 0.702 I. o.7Z- I Q0750 0. 775 - - - f Q 01800 A 0. K 8 0 ? 4 1-*-~ 08~5Q Q~B76 V.1 -- .- I! -- I...--. D3 Q00 0446 O iB APPENDIX C QEras.Pk Yxi& (05 (3) c f0e Cflit(n 017f S- 2 QO LO '*1 TI I.._ r - -t 4= -i ~jt7 4- - I -.7 J.06 090 -. -1- - --- -. CI pt kI Q5~ Ir 4 , --. 0so 1.00 (Y' fe a - 4 - -~ 04008 * I 0480. 0.70 -i 2 04 o ois 0.70 -7 - oe ----- -j0 -0.vt I I -4t4-A D41O . _ t. I- K ..... - QLAID .. ~- -~ ~ .~- :.~. -~ agso. 1T~ V.. 0.10. 02 a04. ~1 1.00 a _ _ _ _ __s 1 43 ;o . ito Vo jwl; I * 00L OL "' 01" Not- - - - -ib I - i lpl 09*0 -AT -1 t -L' T- ~8 Ki~~ QOS t 75 uo.~ Q.0-70 .A. D~Iw - ----1 - IL 1,00 H-- + 0. ( - I-D770 1.00 .4 U---. - I - ~...---.--.--- iA -I - 0.9an 1.00 -, -- ~-~---I- E a.1 0 Q2 0,6 0.4 dr 0,S (.9 4.. L K. 4 4.- d -. I -.. -I -.1'41 I 4. 4- I.. IJ~ -~ -L 1- ;;.~I. t '4 I- 4 1-> I ----K~ 1 4,.74 I.......I 4 7pmI 4 4, -K. ~1-v ,II ~ J~l. ~' J~I ~~~-~ .1 --- 4 7+ ~ -. a -~+ ~.- 10 i~.. VK.0 4 - 7) -. 4.. Cil-4 :4 - -. ,1 4 .4 4 -4 '1 j I I- ~ 1 -4- If t 4 1 I 1 . 1.. T 4 I- 4 .1 1-..' 1 1 r-~- 4. .4 4 1- II 1 I t . . . j . . .i>L'..:... 174 I >4. ..b.v -~-..~---.--,.-. ~1~ 4 L *~. ~.. 4 4 . I A .4 .4 4 70 APPENDIX D (1) Computations for the optimum curves for circular and square tubes, t It a 1/3 1 and d 1/202 1 For square and circular tubes &V = 1/k .qL(20) 80 b+u1 2... ..... For square and circular tubes -y [/k t 0.0406+ .0201 d qLA L / z .k. 1/202 1 . . . . . . (21) Note that equations (20) and (21) are just equation (19) with the appropriate B value substituted in. 71 B 2 t ~1 (1) (2) '14 qai '1 kj (3) (4) 2 V1B)\J()k -1 (3)/k 1 -. 1 (A1)q A 2/A1 -1 (4) 1 B +2B AV/IA1 (2 1 0.24 0.25 0.26 0.945 0.948 0.949 80.8930 80.8987 80.9006 8.9941 8.9944 8.9945 8..5175 8.4877 8.4778 0,06468 0.06096 0.05972 0.0155 0.0152 0.0155 0.09666 0.09930 0.0348 0.0347 * P/Por -0.975 0.36 0.35 0.920 0.918 80.8464 80.8427 8.9915 8.9913 8.7733 8.7944 0.34 0.916 80.8391 8.9911 8.8156 0.10195 0.0346 0.33 0.913 80.8336 8.9908 8.8475 0.10593 0.0349 80.7885 80.7832 80,7779 80.7744 80.7691 80.7621 80.7569 8.9882 8.9880 8.9877 8.9875 8.9872 8.9868 8.9865 9.12a8 9.1559 9.1901 9.2130 9.2476 9.2941 9.3293 0.1402 0.1445 0.1488 0.1516 0.15595 0.16176 0.16616 0.0532 0.0535 0.0536 0.0530 0.053023 0.053381 0.053171 10.0066 9.9401 9,8744 0.2508 0.2425 0.2343 0.07775 0.07760 0.07732 0.07691 * P/Oor =0.950 0.38 0.37 0.36 0.35 0.34 0.33 0.32 0.888 0.885 0.882 0.880 0.877 0.873 0.870 - * P/Por =0.925 0.816 0.821 0.826 80w6659 80.6740 80.6823 8.9814 8.9819 8.9823 0.34 0.831 80.6906 8.9828 9,8096 0.2262 0.35 0.835 80.6972 8.9832 9.7583 0.2198 0.07693 0.36 0.839 80.7039 8.9835 9.7073 0.21341 0.07683 0.37 0.38 0.39 0.40 0.842 0.846 0.849 0,851 80.7090 80.7157 80.7208 80.7242 8.9838 8.9842 8.9845 8.9847 9.6695 9.6196 9.5824 9.5578 0,20868 0.20245 0.1978 0.1947 0.07721 0.076931 0.0771 0.0778 P/Por = 0.875 0.41 0.816 80.6659 8.9814 10.0066 0.2508 0.1028 0.42 0.820 80.6724 8.9818 9.9534 0.24417 0.10255 0.43 0.824 80..6790 8.9822 9.9007 0.23758 0.10216 0.44 0.45 0.46 0.828 0.830 0.833 80.6856 80.6889 80,6939 8.9825 8.9827 8.9830 9.8484 9.8225 9.7839 0.23105 0.22781 0.22298 0.10166 0.1025 0.10257 * Minimum value as indicated by computations * 0.31 0.32 0.33 * P/Por U0.900 72 B m2 t 1 (.1) q k P/Per 0.850 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 P/P 4_ (1+B) -1 (1)+k 0.73 0.779 0.783 0.788 0.791 0.795 0.798 0.800 - (2) 80 (3) 2 (4) Fr2 ) (3)/k -1 A 2 /A 1 .1 /AV/L (4) A2.1 2A+2B)A B +2B 1 80,5975 80.6068 80.6131 80.6219 80.6257 80,6320 80.6368 80.6400 8.9776 8.9781 8.9785 8.9789 8.9792 8v9795 8.9798 8.9800 10.6139 10.5251 10.4667 10.3945 10.3517 10.2949 10.2528 10.2250 0.3267 0.3156 0.3083 0.2993 0.2939 0.2868 0.2816 0.2781 013068 0.12940 0.12949 0.12870 0.12932 0.12897 0.12954 0.13071 10.9676 10.9045 10,8266 10.7804 -10.7193 10.6589 10.5990 10.5398 0.3709 0.3630 0.3533 0.3475 0,3399 0.3323 0.3248 0.3174 0.15949 0.15972 0.158985 0.15985 0.15975 0.15950.15915 0.15870 0.825 0.43 0.44 0.45 0.46 9.47 0.48 0.49 0.50 0.750 0.754 0.759 0.762 0.766 0.770 0.774 0.778 80.5625 80,5685 80.5761 80.5806 80,5868 80e5929 80.5991 80.6053 S.9757 8.9760 8s9764 8.9767 8.9770 8.9774 8.9777 8.9780 0.51 0.780 80.6084 8.9782 10.5105 0.3138 0.160038 0.52 0.783 80.6131 8.9785 10.4667 0.3083 0.160316 80.5300 80.5373 80.5446 80.5506 80.5565 80.5610 80.5670 80.5715 80.5761 80.5806 8.9739 8.9743 8.9747 8,9750 8.9753 8.9756 8.9759 8.9762 8.9764 8.9767 11.3267 11.2432 11.1608 11.0956 11.0312 10.9834 10.9201 10.8732 10.8266 10.7804 0.4158 0.4054 0.3951 0.3869 0.3789 0.3729 0.3650 0,3591 0.3533 0*3475 0.191268 0.190538 0.189648 0.189581 0.189450 0.190178 0.189800 0.190323 0.190782 0.191125 P/Per= 0.800 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0,54 0.55 0.728 0.733 0.738 0.742 0.746 0.749 0.753 0.756 0.759 0,762 x * Minimum value indicated by computations 73 2 B t=1 (1) (2) (3) (4) 4 q k P/Por 0.775 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 (4) (1+B) -1 (1)+k' 0.705 0.710 0.714 0.718 0.722 0.726 0.730 0.733 X 2/A1 -l 80.4970 80.5041 80.5098 80.5155 80.5213 80.5271 80.6329 80.5373 Vfi) (3)/k -1 8.9720 89724 8.9727 8.9730 8.9734 8.9737 8.9740 8.9743 11.7262 11.6371 11.5668 11.4972 11.4285 11.3604 11.2931 11.2432 V/IAl A2 -l (3 +2B3 -1) q 0.4657 0.4546 0.4458 0.4371 0.4285 0.4200 0.4116 0.4054 0.228193 0.227300 0.227358 0.227292 0.227105 0.22680 0.22638 0.227024 0.57 0.736 80.5417 8.9745 11.1936 0.3992 0,227544 0.58 0.739 80.5461 849748 11.1445 0.3930 0.22794 80.4624 80.4679 80.4747 80.4802 80.4858 8,9701 8.9704 8.9708 8.9711 8.9714 12.1913 12.1146 12.0200 11.9453 11.8714 0.5239 0.5143 0.5025 0,4931 0.4839 0.26719 0.26744 0.26632 0.26627 0.266145 0.26588 0.51 0.52 0.53 0.54 0.55 0,680 0.684 0.689 0.693 09.697 0.56 0.701 80,4814 8.9719 11.7984 0.4748 0.57 0.703 80.4942 8.9721 11.7623 0.4702 0.26801 0.58 0.706 80.4984 8.9723 11.7083 0.4635 0.26883 P/Por 0.725 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.655 0.660 0.664 0.669 0.672 0.677 0.680 0,683 80.4290 80.4356 80.4409 80.4476 80.4516 80.4583 80,4624 80.4665 8.9682 8.9686 8,9689 8.9693 8.9695 8.9699 8w.9701 8.9703 12.6919 12.5887 12.5073 12.4070 12.3474 12.2494 12.1913 12.1336 0.5864 0.5735 0.5634 0.5508 0.5434 0.5311 0.5239 0.5167 0,310792 0.309690 -0.309870 0.308448 0-309738 0.308038 0.309101 0.31002 0.61 0.62 0.687 0.690 80.4720 80.4761 8.9706 8.9709 12.0576 12.0013 0.5072 0.5001 0.30939 0.31006 * Minimum value indicated by computations * P/Por a0.750 ,-4 74 B 2 t 1 (1) q k 4 (1+B) .1 (2) (1)+k2 (3) (4) q (3)/k -1 A 2/Al -1 /1 B2+2B A2 (-1) A 0.6182 0.6048 0.5943 0.5838 0.5761 0.35237 0.35078 0.35064 0.35028 0.35142 (4)2 - 1 1 P/Porn 0.700 0.57 0.58 0.59 0.60 0.61 0,643 0.648 0.652 0.656 0.659 P/Pcr 8094134 80w4199 8094251 80.4303 80.4343 8.9674 8.9677 8o9680 8.9683" 8.9685 12.9461 12.8390 12.7546 12.6711 12.6092 0.675 0.59 .0.619 80.3832 8.9657 13.4841 0.6855 0.404445 0.60 0.61 0.62 0.63 0.64 0.623 0.627 0,631 0.634 0.638 80.3881 80.3931 80.3982 80.4020 80.4070 8.9659 8.9662 8.9665 8.9667 8.9670 13.3914 13.3001 13.2099 13.1430 13.0548 0.6739 0.6625 0.6512 0.6428 0.6318 0.404340 0.404125 0.403744 0.404964 0.404352 0.65 0.66 0.640 0.643 80.4096 80.4134 8.9671 8.9674 13.0110 12.9461 0.6263 0.6182 0.407095 0,408012 0.67 0.68 0.647 0.650 80.4186 80.4225 8.9676 8.9679 12.8602 12.7967 0.6075 0.5995 0.407025 0.407660 80.3352 80.3411 80.3469 80.3516 80.3564 80.3612 80o3660 80.3709 80.3733 80.3770 8.9630 8.9633 8.9636 8.9639 8.9642 8.9644 8.9647 8.9650 8.9651 8.9655 14,4801 14.3481 14.2183 14.1161 14.0164 13.9158 13,8176 13.7208 13.6728 13.6014 0.8100 0.7933 0.7772 0.7645 0.7519 0.7394 0.7272 0.7151 0.7091 0.7001 0.4617 0.460114 0.458548 0.4587 0.458659 0.458428 0.458136 0.457664 0.460915 0.462066 P/Por 0.57 0.58 0.59 0.60 0.61 0.62 0;63 0.64 0.65 0.66 0.650 0.579 0.584 0,589 0.593 .597 0.601 0.605 0.609 0.611 0.614 * Minimum value indicated by computations q 75 =2 t 1 (1) q k 2 (1+B) .1 (1)+k' P/Por a0.625 0.61 0.566 0.62 0.570 0.63 0.575 0.64 0.579 0.65 0.66 0.585 0,587 0.67 0,590 0.68 0.593 0.69 0.70 (2) 80.3204 80.3249 80.3306 80,3352 80.3399 80.3446 80.3481 (3) V(2) 8. 9622 8.9624 8.9627 809630 0.596 80.3516 80.3552 8.9633 8.9635 8.9637 8,9639 8.9641 0.599 80.3588 8.9643 P/Por a0.600 0.65 0.553 0.66 0.558 0.67 0.561 0.68 0.565 0.69 0,569 0.70 0.571 P/Por =0.575 0.63 0.516 0.64 0.65 0.520 0.525 0.66 0.530 0.67 0.533 0.68 0.69 0.537 0.540 0.70 0.71 0.72 0.543 0.546 0.550 P/Pora0.550 0.67 0.505 0.68 0.510 0.69 0.513 0.70 0.517 80.3058 8.9614 80.3114 8.9617 8.96185 8.8621 80.3147 80.3192 80.3237 80.3260 80.2662 80.2704 80.2756 80.2809 80.2840 80*2884 80.2916 80.2948 80.2981 80.3025 (4) (3)/k -1 14.8372 14.7235 14.5873 14.4801 14.3744 14.2700 14.1927 14.1161 A 2/A.1 bV/L% (4)A2 -1%( 2 1) q 1 B +2B 0.8542 0.8404 0.8234 0.8100 0.7968 0.7645 0.7550 0.51986 0.52095 0.7456 0.52192 14.9747 0.9006 0.8825 0.8718 14.0404 13.9654 15.2050 15.0603 14.8621 0.8577 8.96235 8.9625 14.7510 14.6961 0,8438 0.8370 0.585390 0.582245 0.584106 0.583236 0.582222 0.585900 8,9591 8.9594 809597 8.9600 16.3625 1.0453 1.0287 0.658539 0.658368 1.0082 0.9945 0.655330 0.656370 8.9601 15.8106 15.6860 15.5937 15.5022 15.4119 15. 2930 0.9763 0.9607 0.9492 0.9377 0.654121 0.653276 0.9264 0,657744 009116 0.656352 1.0924 0.731908 0.728076 0.729951 0.728770 0.734688 809604 8.9606 8.9607 8.9609 8.9612 16.2296 16.0660 15.9056 80*2601 80.2632 8.9590 80,2673 8.9592 16.3292 8.9594 8.9595 16.2296 1.0411 1.0287 16.1637 1.0204 0.520 80.2704 0.72 0.522 80.2725 0.51840 0.7740 0.7837 16.7396 16.5662 16.4639 0.71 0.51874 0.51792 0.51724 0.51858 8.9585 8,9588 80.2550 0.52106 0.52104 * Minimum value indicated by computations 1.0707 1.0539 0.654948 0,65639 0.730377 * B 76 t 1 B (1) q k (1+B)4 -1 (2) (3) (1)+k2 (4) A 2/A1 (3)/k -1 .0406 .1 &V/Ikl -1 (A -1) B+2B 1 0.0291 0.0282 0.942 0.945 0.25 0.26 0.27 0.9280 0.9336 0.96333 0.96623 0.02264 0.02246 0.1264 0.1174- 0.948 0.9393 0.96918 0.02234 0.1111 0.0278 0.949 0.950 0.9412 0.9431 0.97016 0.97113 0.02229 0.02224 0.1089 0.1065 0.0283 0.0297 P/Por= 0.950 0.36 0.920 0.35 0.918 0.34 0.916 0.8870 0.8833 0.9797 0.946181 0.93984 0.93792 0.02370 0.02379 0.02393 0.1791 0.1836 0.1905 0.6448 0.6426 0.6477 0.33 0.9742 0.93499 0.02408 0.1980 0.6534 0.913 0.39 0.889 0.8309 0,91154 0.02535 0.2612 001019 0.38 0.888 0.8291 0.91055 0.02539 0.2632 0.1000 0.37 0.36 0.35 0.34 0.885 0.882 0.880 0.877 0.8288 0.8185 0.8150 0.8097 0.90763 0.90471 0.90277 0.89983 0.02557 0.02574 0,02587 0.02604 0.2721 0.2806 0.2871 0 .2950 0.1006 0.1010 0.1005 0.1003 0.33 0.873 0,8027 0.89594 0.02627 0.3070 0.1013 0.32 0.870 0.7975 0.89303 0.02647 0.3169 0.1014 P/Per 0.900 0.31 0.816 0.7065 0.84054 0.03007 0.4960 0.1538 0.32 0.33 0.34 0.35 0.821 0.826 0.831 0.835 0.7146 0.7229 0.7312 0.7378 0.84534 0.85024 0.8551 0.85895 0.02964 0.02934 0.02900 0.02868 0.4746 0.4597 0.4428 0.4269 0.1519 0.1517 0.1506 0.1494 0.36 0.839 0.7445 0.86284 0.02841 0.4134 0.1488 0,37 0.38 0.39 0.40 0..842 0.846 0.849 0.851 0.7496 0.7563 0.7614 0.7648 0.86579 0.86966 0.87258 0.87453 0.02825 0.02796 0.02777 0.02764 0,4055 0.3910 0.3816 0.3751 0.1500 0.1486 0.1488 0.1500 P/Por * 0.925 * cr 0.875 0.41 0.816 0.7065 0.84054 0.03007 0.4960 0.2034 0.42 0.43 0.44 0.45 0.46 0.47 0.820 0.824 0.828 0.830 0.833 0.835 0.7130 0.7196 0.7262 0.7295 0.7345 0.7378 0.84439 0.84829 0.85217 0.85411 0.85703 0.85895 0.02974 0.02947 0.02919 0.02904 0.02884 0.02808 0.4796 0.4662 0.4522 0.4448 0.4348 0.4269 0.2014 0.2005 0.1990 0.2002 0.2000 0.2006 * Minimum value indicated by computations * P/P o * 0.23 0.24 * P/P rx 0.975 77 t n B q k (2) (1+B) 4 1 (1)+k .0406 (3) 4(2) (4) (3)/k -1 A 2/ /A 1j. Av/L1 a74- * (- 1) q .- B2 P/Por* 0.850 0.40 0.773 0.6381 0.79881 0.03338 0.6607 0.2643 0.41 0.42 0.779 0.783 0.6474 0.6537 0.80461 0.80852 0.03287 0.05259 0.6353 0.6214 0.2605 0.2610 0.43 0.44 0.45 0.46 0.788 0.791 0.795 0.798 0.6615 0.6663 0.6726 0.6774 0.81333 0.81627 0.82012 0.82304 0.03214 0.03194 0.03159 0.03137 0.5990 0.5891 0.5716 0.5607 0.2576 0.2592 0.2572 0.2579 0.47 0.800 0.6806 0.82498 0.03122 0.5532 0.2600 0.03456 0.03507 0.7642 0.7448 0.3286 0.3277 0.3258 * (1) 0.43 0.44 0.750 0.754 0.6031 0.6091 0.77660 0.78045 0.45 0.759 0.6167 0.78530 0.03465 0.7239 0.46 0.762 0.6212 0.78816 0.03433 0.7080 0.3257 0.47 0.766 0.6274 0.79209 0.03406 0.6945 0.3264 0.48 0.770 0.6335 0.79593 0.03367 0.6751 0.3240 0.49 0.50 0.774 0.778 0.6397 0.6459 0.79981 0.80368 0.03334 0,03300 0.6587 0.6418 0.3228 0.3209 0.51 0.52 0.53 0.54 0.780 0.783 0.786 0.788 0.6490 0.6537 0.6584 0.6615 0.80561 0.80852 0.81142 0.81333 0.03283 0.03259 0.03234 0.03214 0.6333 0.6214 0.6090 0.5990 0.3230 0.3231 0.3228 0.3235 or 0.800 0.46 0.47 0.728 0.783 0.5706 0.5779 0.75538 0.76020 0.03760 0.03710 0.8706 0.8458 0.4005 0.3975 0.48 0.49 0.738 0.742 0.5852 0.5912 0.76498 0.76890 0.03655 0.03625 0.8184 0.8035 0.3928 0,3937 0.50 0.746 0.5971. 0.77272 0.03581 0.7816 0.3908 0.51 0.749 0.6016 0.77563 0.03555 097687 0.3920 0.52 0.53 0.753 0.756 0.6076 0.6121 0*77949 0.78237 0.03517 0.03488 0.7498 0.7353 0.3899 0.3897 0.54 0.759 0.6167 0.78530 0.03465 0.7239 0.3909 0.55 0.762 0.6212 0.78816 0.03433 0.7080 0..3894 0.56 0.57 0.58 0.59 0.764 0.767 0.770 0.772 0.6243 0.6289 0.6335 0.6366 0.79013 0.79303 0.79593 0.79787 0.03420 0.03393 0.03367 0.03351 0.7015 0.6881 0,6751 0.6672 0.3928 0.3922 0.3916 0.3936 * Minimum value indicated by computations * P/P o * P/P r- 0.825 f7 ? t l B 202 B=1 (1) (1+B) 4 -1 k (1)+k2 (3) 2 ( (4) 4(2-(3)k (3)/k -1 A V/IAI A (4) B +2B 0.0406 P/Por= 0.775 0*49 0.705 0.710 0.50 0.714 0.51 0.718 0.52 0.722 0.53 0.54 0.726 0.55 0.730 0.733 0.56 0.736 0.57 0.739 0.58 0.741 0.59 0.60 0.743 A2 .1 I1 0.5376 0.5447 0.5504 0.5561 0.5619 0.5677 0.5735 0.5779 0.5823 0.5867 0.5897 0.5926 0.73321 0.73804 0.74189 0.74572 0.74960 0.75346 0.75730 0.76020 0.76309 0.76596 0.76792 0.76981 0.04001 0.03949 0.03906 0603860 0.03822 0.03782 0.03739 0.03710 0.03680 0.03648 0.03632 0.03608 0.9905 0.9647 0.9433 0.9204 0.9015 0.8816 0.8602 0.8458 0.8308 0.8149 0.8070 0.7950 0.4853 0.4824 0.4811 0.4786 0.4778 0.4761 0.4731 0.4736 0.4736 0.4726 0.4761 0.4770 0.70923 0.71309 0.71784 0.72166 0.72553 0.72938 0.73130 0.04298 0.04252 0.04185 0.04135 0.04093 0,04048 0.04025 1.1383 1.1154 1.0821 1.0572 1.0363 1.0139 1.0025 0.5805 0.5800 0.5735 0.5709 0.5700 0.5678 0.5714 P/Por 0.51 0.52 0.53 0.54 0.55 0.56 0.57 = 0.750 0.680 0.684 0.689 0.693 0.697 0.701 0.703 0.5030 0.5085 0.5153 0.5208 0.5264 0.5320 0.5348 0.58 0.706 0.5390 0.73410 0603980 0.9801 0.5685 0.59 0.709 0.5433 0.73709 0.03961 0.9706 0.5727 0.60 0.711 0.5461 0.43989 0.03935 0.9577 0.5746 P/Por 0.725 0.53 0.54 0.55 0.56 0.655 0.660 0.664 0.669 0.4696 0.4762 0.4815 0.4882 0.68527 0.69007 0.69390 0.69871 0.04621 0.04556 0.04503 0.04440 1.2990 1.2667 1.2403 1.2090 0.6885 0.6840 0.6821 0.6770 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.672 0.677 0.680 0.683 0.687 0.690 0.692 0.695 0.4922 0.4989 0.5030 0.5071 0.5126 0.5167 0.5195 0.5236 0.70157 0.70633 0.70923 0.71211 0.71596 0.71882 0.72076 0.72360 0.04400 0.04332 0.04298 0.04262 0.04215 0.04176 0.04156 0.04115 1.1891 1.1552 1.1383 1.1204 1.0970 1.0776 1.0677 1.0473 0.6778 0.6700 0.6716 0.6722 0.6692 0.6681 0.6726 0.6703 0.65 0.66 0.698 0.700 0.5278 0.5306 0.72650 0.72842 0.04083 0.04060 1.0313 1.0199 0.6703 0.6731 * q (2) 79 t 1 B 10 (1) (2) (3) 472-(4 q k (1+B) -1 (1)+k (4) A2 A2/L q(2) (3)/k -1 (4 B +2B 'cro 0.700 0.57 0.643 0.58 0.648 0.59 0.652 0.60 0.656 0.61 0.659 0.62 0.661 1 (q 1 P Pl/Por 0.4540 0.4605 0.4657 0.4709 0.4749 0.4775 0.67380 0.67860 0.68242 0.68622 0.68913 0.69101 0.04790 0.04722 0.04665 0.04606 0.04572 0.04540 1.3831 1.3493 1-03209 1.2915 1.2746 1.2587 0.7883 0.7826 0.7793 0.7749 0.7775 0.7804 0.4238 0.4287 0.65100 0.65475 0.05169 0.05096 1.5716 1.5353 0.9272 0.9212 0.675 0.59 0.60 0.619 0.623 0.61 0.627 0.4337 0.65856 0,05033 1.5040 0.9174 0.62 0.631 0.4388 0.66242 0.04979 1.4771 0.9158 0.63 0.634 0.4426 0.66528 0.04933 1.4542 0.9161 0.64 0.638 0.4476 0.66903 0.04863 1.4194 0.9084 0.65 0.66 0.640 0.643 0.4502 0.4540 0.67097 0.67580 0.04839 0.04790 1.4075 1.3831 0.9149 0.9128 0.67 0.68 0.69 0.647 0.650 0.651 0.4592 0.4631 0.4644 0.67764 0.68051 0.68147 0.04735 0.04693 0.04680 1.3557 1.3348 1.3284 0.9083 0.9077 0.9166 0.70 0.653 0.4670 0.68337 0.04650 1.3134 0.9194 1.0966 1.0910 1.0793 P1/Pcr= 0.650 0.57 0.58 0.59 0.579 0.584 0.589 0.3758 0.3817 0.3875 0.61303 0.61782 0.62250 0.05877 0.05791 0.05687 1.9239 1.8811 1.8294 0.60 0.593 0.3922 0.62626 0.05608 1.7900 1.0740 0.61 0.62 0.597 0.601 0.3970 0.4018 0.63008 0.63388 0.05541 0.05470 1.7567 1.7214 1.0716 1.0673 0.63 0.605 0.4066 0.63765 0.05396 1.6846 1.0613 0.64 0.65 0.609 0.611 0.4115 0.4139 0.64148 0.64335 0.05333 0.05294 1.6532 1.6338 1.0580 1.0620 0.66 0.614 0.4176 0.64622 0.05247 1.6104 1.0629 0.67 0.617 0.4213 0.64908 0.05199 1.5866 1.0630 0.68 0.619 0.4238 0.65100 0.05169 1.5716 1.0687 80 1 t 1 1 B 100 (1) (2) (3) (4) A2 / 1 q k (1+B) -. 1 (1)+k \J(2) 1)q (3)/k-1 B2+2B 1 P /Por= 0.625 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.566 0.570 0.575 0.579 0.583 0.587 0.590 0.593 0.596 0.599 0.3610 0.3655 0.3712 0.3758 0.3805 0.3852 0.3887 0.3922 0.3958 0.5994 0.60083 0.60457 0.60926 0.61303 0.61685 0.62064 0.62346 0.62626 0.62913 0.63198 0.06153 0.06064 0.05958 0.05877 0.05806 0.05731 0.05671 0.05608 0.05558 0.05505 2.0612 2.0169 1.9642 1.9349 1.8886 1.8512 1.8214 1.7900 1.7652 1.7388' 1.2573 1.2505 1.2374 1.2313 1.2276 1.2218 1.2203 1.2172 1.2180 1.2172 0.71 0.72 0.601 0.603 0.4018 0.4042 0.63388 0.63577 0.05470 0.05434 1.7214 1.7035 1.2222 1.2265 0.73 0.606 0.4078 0.63859 0.05377 1.6751 1.2228 0.74 0.609 0.4115 0.64148 0.05333 1.6532 1.2234 Pl/Pr = 0.600 I r 0.65 0.553 0.3464 0.58856 0.06430 2.1990 1.4294 0.66 0,558 0.3520 0.59330 0.06326 2.1473 1.4172 0.67 0.561 0.3555 0.59624 0.06281 2.1249 1.4237 0.68 0.565 0.3598 0.59983 0.06164 2.0667 1.4054 0.69 0.70 0.569 0.571 0.3643 0.3666 0.60357 0.60548 0.06075 0.06038 2.0224 2.0040 1.3955 1.4028 0.71 0.573 0.3689 0.60737 0.05998 1.9841 1.4087 0.72 0.576 0.3724 0.61025 0.05946 1.9582 1.4099 0.73 0.74 0.75 0.579 0.580 0.581 0.3758 0.3770 0.3782 0.61303 0.61400 0.61498 0.05877 0.05862 0.05848 1.9239 1.9164 1.9095 1.4044 1.4181 1.4321 81 t 1 B 10 (1) (2) (3) (4) A2 -1 q k 4 (1+B) -1 2 (3/A (3)/k-1 (1) +k (4- hYIA -1 ( 21 -1) q P /'cra 0.575 0.71 0.72 0.73 0.74 P 'r 00540 0.543 0.546 0.550 0.551 0.553 0.70 0.71 0.-72 0.73 0.74 0.522 0.525 0.527 0.68 0069 0*07108 0..06983 2.5363 204741 1.6486 1.6329 0.06893 2.4294 1.6277 0.3290 0.3322 0.3354 0.57359 2.3896 2.3507 0.57914 0.06813 0.06735 0.06655 0.3387 0.3431 0.58198 0.58575 0.06589 0.06500 0.3442 0.3464 0.58669 0.58856 0.06477 0.54369 0.07661 0.07521 0.07442 0.07328 0.3215 0.3246 0.57637 0.06430 2.3109 2. 2781 2.2338 2.2224 2.1990 1.6249 1.6220 1.6176 1.6175 106083 1.6223 1.6273 m0.550 0.505 0.510 0.513 0.517 00520 0.67 0.56232 0.56701 0.56974 0.3162 * 0.67 0.68 0.69 0.070 0.525 0.530 00533 0.537 0.2956 0.3007 0. 3038 0.3079 0.3110 0 3131 0.54836 0.55118 0;6 55489 0.55767 0.55955 0.3162 0.56232 0.3183 0.56418 2.8114 2.7418 2.7025 1.8836 2.6458 2.6040 1.8521 0.07244 0.07193 0.07108 0.07055 2.5786 2. 5363 2.5100 1.8644 1.8647 1.8488 1.8566 1.8515 1.8574 * 0.65 0.66 Zt 0. c -_j 7t2 '2 -- --'44. u-b -T4- K.-7 .uj -.1 4 f -'- -~it5PU 7 F XiL4 Ii' -4 F F F i 4 t - 44 F., Y -4 -J -F' 7-I - J ~i1 -F F i4~r F 4-F. '4- 1 - - -; [7 '4., 1. ,T F j2~V 'F FF 1 17- Iii ii] 7Y7 PT. L~t ~F;4-i- -F 4 - 'F ''1 'a VI. 1---'-; i: 1i21.221 4- 4- i-F--F 3- -1 -4. H-!~4I 4- .27 I 1-t1 -fl-i-I 14- K 44;. -4--f---FF4 4+ l ~iii:TiIPi j 4'~7ti 17 7j-77 ' 4i k7 '4~ v~ J t F.~FIF- 4-7 77 :4*77' ~ :--7 4FF .4 F -FT *T F 4Ft~-T t4 7 -- 47 444 , At ' F' F F --~. F' -F.-L - F 4 , , F. F> F 4 I FFF 1 4 IF [F.:t V7 r .''..4:7. ' ~F - F - F7, '4 SI, F, ; 7I~ 17 -F F .4. F 4~F.F4~ '.4. '--4.' 1. F "'[ii Ft I44 -44 1-., - I-: - F 7t1'J F - F -F 4*' : ''F~''F I- 1 1 t 71 _ 1 ZTI 1 4- :4-I ' 'F' 1.4 0o- ,I~ -] ac 83 APPENDIX E Algebra of method for checking validity of equation (26) A J+ J + H where H * B 4c = 1 0 0 . 0 0 (2D - 1) B or d 1 d 1 t . . . . . . . . H . 2 0 [ j .- ..J - 1 1 d o g1 1c- B 1/2" -2(B1+ 1) ' 2(BB+ 1) b but special condition D D 1 - c0 .0 0 0 0 0 0 (26) d4 d 1d * 1 o d 0 d1 D W B +1 Dc 2 402 = (B+1) B 4 . . . . (26a) (2D - 1) 4c . 0 2 e8Dc - 4c - = 0 k12G k2 B 2(B + 11 B 2(B+ 1) 2. B 2(B + 1) 2 B TB + 1)r B 4(B + 1) 0 0 0. 0 . (26b) 84 B2 B2 4c2 (2D - 1)- 2B +2 (B + 1) M1) B 4 2(2D (2B + 2 B) - B(B + 2) (B + 1) - (B + 1) G defined by equation 2(1-2cD) 03 (23a) .2c(1 1 o 2cD n 1 - 2 r Z(B+B17 2B B+ - 2cD) 3 0o B +1 )4 (1.. G - 1 Ci 12B+f A2 3 - (I - 2c)4] - 6(l .. 2cD) c (,.C)2 G = 11 B2 (1. 1 ) + v[1 (B+1) B S )4][ B+1 A1 12 . +1 k B(B + 2) (B + 1) multiply top and bottom by (B A2 [2B(B+1) ~ B2 - (B+1) 2 + 1) 2 - 1 ) + q[(B+1) 10 + (B+1)4 i1 2 B 2+ 2B 2 -I4 0 1 + 1/k v(B+1) . 1] l - k2 + (B+1) 42'; k2 B2 + 2B A r~2 1 1/k (B+1) 2 B + 2B Equation (26) checks* + k2 * # 0 0 0* 0 * (18) 85 APPENDIX F Computations for spot check of the conclusions drawn from Table II. &V - 558.03 [ \ 0.0051469 ( 1 . - k 1 ) + 1 1 3 qL A *.(29) 1 D m4 k q (A) 558.03 (A) AV Trw 1r- 0.750 0.53 0.689 0.002922 106306 0.8642 0.54 0.69,3 0.002781 1.5519 0.8380 0*55 0.697 0.002740 1.5290 0.8410 0.56 0.701 0.002660 1.4844 0.8312 0.:57 0.703 0.002630 1.4676 0.8365 0.58 0.706 0.002586 1.4430 0.8370 0.59 0.709 0.002543 1.4191 0.8373 0.60 0.711 0.002514 1.4029 0.8417 I (A) - 1 0.0051469 (kya + 1 ) Notet - 1 ] * cr W qq-wl- 86 APPENDIX G Computation of minimization criteria for circular and square tubes* (See Table I.) G(kB) o k[(1+B) 4 14 (1+B) - 1(22a) k n 1.00 G(k,B) = (B+l) 2 k 2(1+B)401+k2 1/2 B I 2 1 (1+B) 4 - 1 4 G(kB) = 0 k = 1.00 G(kB) = k =*90 B 0 B = 2 G(kB) = 1 Eo.9 (80.81) - 9 x 0.81 x 8.98944] G(kB) a 0.08995 k = 0.90 B - [(2+B) 44(B+1)2 (B+)41/2 1 + k2 ] G(kB) - 24.6281 [0.9 (0.85060401 - 1.0201 x 0.81 (0.92228)] = 24.6281 L0.765543609 - 0.762062441] 24.6281 [Q.00348117] G(kB)= 0.08573 k = 0.70 B = 2 G(kB)- [0.7 (80.49) - 9 x 0.49 x 8.9716] [ [56.34300 - 39.564756] - G(kB)m ( 16 778244] 0.20972 87 B m k a 0.70 16 G(k,B) - 24.6281 [ 0.7 (0.53060401)- 1.0201 x 0.49 (0.72843)] = 24.6281 f 0.3714228 - 0.3641050] = 24.6281 f 0.0073178] G(kB) n 0.18022 k n 0.50 B = 2 G(kB) - [ 0.5 (80.25) . 9 x 0.25 x 8.95824)] 40.12500 - 20.15604] 80 G(kB) - k = 0.50 [19*96896] 0.24961 B G(k,B) n 24.6281 [0.5 (0.29060401) - 1.0201 x 0.25 (0.53908)] = 24.6281 [0.145302005 - 0.137478877] = 24.6281 [0.00782313] G(kB) a 0.19267 8e APPENDIX H Computations for minimization criteria for structural cross section. (See Table I.) + 12G 1 g(kc,D) 11 + 1] . . (28a) kk 1 12 4 6c(1..2cD)(1-c) 2-2(1*-20D)c 3 .20(1-2cD) 1~-- (1--2c) .- t (1) C 1 t a= (2) 4 G 20 G = 1 [0.18549371 D. 4 G = 1 tO.0051469] D 20 G = [0.025 1289] D - G D * (23a) T2 [0.0457937] =1 t (3) al= C W3 = 1 t (4) -= k = 1.00 (1) g(k,cD) = 0 (2) g(k,c,D) = 0 (3) g(kcD) a 0 (4) g(k,cD) a 0 k = 0.90 3 (1) [0.0457937 x 0.234567 + 1 - g(k,c,D) = Q.729 [1.01074169 - 1.0052 - - am 0.0457937 g(kcD) a 0.08593 * I ] 89 0. g(kc,D) Q*3 3. 0.729 I [0.1854937 x 0.234567 + 1 . [1.0435107 - 1.0211 . (2) g(k,c,D) - 0.08650 (3) g(kc,D) = 0.0519 1.0012072928923 = 141.63865 [1.0012073 ] V 1.0006035) g(k,c,D) a 0.08552 (4) g(k~c*D) n 0,729 [90912 0.0253289 [1.005941324 .. n 0729 I [1.0059413 - 1.0029638] g(kc,D) - 0.08569 k = 0.70 (1) g(kpD) 0.43 [1.047662816 0.057937 1.023551 g(kcD) a 0.18061 (2) (3) (4) 0.343 g(k,c,D) ; g(k,c,D) U 0.18642 g(kocD) U 0.,343 g(kcD) U 0.17873 g(k,c,D) U g(kcD) U 0 0.17960 [1.19306481 . 1.092251 [1.005357 - 1.0026753 [1.0263627 - 1.0131 - - I -4 90 k n 0.50 g(kcD) = 0.1 [1.1373811 0.0457937 - (1) 1.0665 " - I g(kocD) = 0.19348 - 0.*125 g(kcD) m 0.15 [ 1.556481 1.2476 - - I 10154407 1.0077 - - I - (2) g(k,cD) n 0.20815 (3) g(k,c,D) n 0.0051 69 g(k,c,D) n 0.18799 (4) 0.,125 g(kcD) g(kcpD) - 0.19092 [1.07 9867- 1.03730 -- I 91 BTBLI OGRA.PHY American Institute of Steel Construction, Steel Construction Manual, Fifth Edition, (1947) New York, N. Y. Bleich, Friedrich, The Buckling Strength of Metal Structures, Critical Survey, Contract NObs-45424, A David Taylor Model Basin, Bureau of Ships, Navy Department, with Frankland and Lienhard. Dinnik, A. N., (1949) Design of Columns of Varying Cross Section, Trans. actions A.S.M.E., Applied Mechanics Division, Volume 54 (1932) pp. 165 - 171. Formed Steel Tube Institute, Handbook of Welded Steel Tubing. 1621 Euclid Avenue, Cleveland, Ohio. (January 15, 1941.) Summerill Tubing Company, Bridgeport, Conn. Timoshenko, S. , Summerill Aircraft Tubing Data, (March 8, 1943.) Theory of Elastic Stability, Company, Inc., New York, N. Y. (1936) McGraw-Hill Book