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INSTABILITY OF CYLINDRICAL SHELLS WITH INCLINED STIFFENERS By RU LIN LEE A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1965 ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to all the members of his supervisory committee. In particular, he wishes to thank Dr. S. Y. Lu, Chairman of his supervisory committee, for constant advice and encouragement throughout the entire period of this research. He would also like to thank Dr. William A. Nash, Chairman, Department of Engineer ing Science and Mechanics, for his valuable suggestions and financial support throughout the author's entire graduate study program. He is also indebted to Dr. I. K. Ebcioglu, Department of Engineering Science and Mechanics, Dr. J. Siekmann, De partment of Engineering Science and Mechanics, and Dr. R. G. Blake, Department of Mathematics, for serving on his supervisory committee and for the various stimulating discussions he has held with them over the past few years. Final thanks go to the National Aeronautics and Space Administration for their sponsorship of this study. TABLE OF CONTENTS ACKNOWLEDGMENTS . LIST OF TABLES . LIST OF FIGURES . LIST OF SYMBOLS . ABSTRACT . CHAPTER * C * * C C C * C C C C C * C C Page S. ii . V . vi S. viii * xi I. INTRODUCTION . . II. BASIC RELATIONS. . . 2.1 Compatibility Equations . 2.2 Energy Expressions .. III. INSTABILITY OF A STIFFENED CYLINDRICAL SHELL UNDER AXIAL COMPRESSION AND INTERNAL PRESSURE . .. . 3.1 Deflection Pattern and Stress Function. 3.2 Expressions of Total Potential Energy . 3.3 Minimization of Total Potential Energy. IV. INSTABILITY OF A STIFFENED CYLINDRICAL SHELL UNDER BENDING AND INTERNAL PRESSURE. . 4.1 Deflection Pattern and Approximate Stress Function . . iii TABLE OF CONTENTS (Continued) 4.2 Expressions of Total Potential Energy . 4.3 Minimization of Total Potential Energy. V. INSTABILITY OF A STIFFENED CYLINDRICAL SHELL UNDER TRANSVERSE LOAD AND INTERNAL PRESSURE. 5.1 Deflection Pattern and Approximate Stress Function . . 5.2 Expressions of Total Potential Energy . 5.3 Minimization of Total Potential Energy. VI. NUMERICAL EXAMPLES . . 6.1 Cylinder under Bending and Internal Pressure or Transverse Shear and Internal Pressure . . 6.2 Cylinder under Axial Compression and Internal Pressure . . VII. EXPERIMENTAL INVESTIGATION . 7.1 Models . . 7.2 Test Result .. . * VIII. DISCUSSIONS AND CONCLUSIONS.. . APPENDIX A . . APPENDIX B . . BIBLIOGRAPHY. .. . BIOGRAPHICAL SKETCH .. . .. 74 . 81 . .. 88 . 93 Page 31 35 40 40 41 43 44 44 46 48 48 50 51 \ LIST OF TABLES Table Page 1. Tests of Stiffened Thin Mylar Cylinders under Axial Compression or Bending without Internal Pressure... .. ..... 54 LIST OF FIGURES Page Figure 1. Coordinates and Displacement Components of a Point on the MiddleSurface of the Shell . 2. The Coordinate System of the Stiffened Shells and Stiffeners . . 3. The Dimensions of the Stiffened Shells under Different Loadings . . * 56 * 57 4. Stress Parameter, 12 as Deflection, n . 5. Stress Parameter, F2 as Deflection, t . 6. Stress Parameter, (2 as Deflection, . 7. Stress Parameter, 2 as Deflection, 7 . 8. Stress Parameter, 92 as Deflection, . 9. Stress Parameter, 52 as Deflection, n . 10. Stress Parameter, 2, T WaveLength Ratio, C . a Function of a Fu n o. a Function of *a Fn of a Function of a Function of a Function of a Function of as a Function * & 0 0 0 0 * 58 * 59 . 60 S. .. 61 .. 0 62 0 10 11. Minimum Stress Parameter, 2, as a Function of Deflection . 12. Minimum Stress Parameter, ~2, 3 as a Function of Deflection n, at Different Inclined Angles .. . vi . 63 * 64 . 65 * * LIST OF FIGURES (Continued) Figure Page 13. Relation Between Minimum Stress Parameter, 12, l/ and Inclined Angles .. 67 14. Relation Between Minimum Stress Parameter, 2,,y aand Rigidity of the Stiffeners. 68 15. Relation Between Minimum Stress Parameter, 32,r ,R and Internal Pressure p 69 16. Comparison of Effect of the Imperfection Ratio on the Stiffened Shells Under Axial Compression 70 17. Typical Buckling Patterns for Unpressurized Stiffened Shell Under Axial Compression 71 18. Comparison of Theoretical and Experimental Results on the Stiffened Shells Under Pure Bending ........ ..... 72 vii LIST OF SYMBOLS Et3 D Flexural rigidity of the shell =12 12(1 92) E Young's modulus for shell Ej, Ek Young's modulus for jth and kth stiffeners, k respectively Ej, E Dimensionless Young's modulus for jth and kth stiffeners, respectively G, Gk Shear modulus for jth and kth stiffeners Gj, G Dimensionless shear modulus for j and k stiffeners, respectively Ij, Ik Moment of inertia of jth and kth stiffeners about center of gravity of jth and kth stiffeners respectively I I Dimensionless moment of inertia of jth and Sk kth stiffeners about center of gravity of jth and kth stiffeners respectively J J Polar moment of inertial of jth and kth j k stiffeners Jj, Jk Dimensionless polar moment of inertial of jth and kth stiffeners K Torsional rigidity of the stiffeners L Length of the shell I Total length of stiffeners m, n Numbers of waves in the axial and circum ferential directions Nk Number of stiffeners in ~1direction viii LIST OF SYMBOLS (Continued) PJ p PC R t tl U, V, W w o x, y, z yI' T'2 Th'~Izl ALe Number of stiffeners in y2direction Transverse load Internal pressure Axial load Dimensionless transverse load Dimensionless internal pressure Radius of the shell Thickness of shell Thickness of stiffeners Components of displacements in x, y, z directions, respectively. Fig. 1 Initial deflection Orthogonal coordinates on median surface of shell Dimensionless parameter defined in Eq. (22) w Imperfection ratio = o The angles between the stiffeners and the generator of the cylinder, respectively Ratio of arbitrary parameters defined in Eq. (48) Ratio of numbers of waves in the circum ferential and axial direction Poisson's ratio for shell Ec' 5ib' 6c b, U Average axial compression stress, bending stress, and eccentric compression stress, respectively Dimensionless axial compression stress, bending stress, and eccentric compression stress, respectively Laplace operator Biharmonic operator = (V2)2 Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INSTABILITY OF CYLINDRICAL SHELLS WITH INCLINED STIFFENERS By RuLin Lee August, 1965 Chairman: Dr. S. Y. Lu Major Department: Engineering Science and Mechanics An analytic study of the general stability of thin circular cylindrical shells stiffened by a system of inclined stiffeners has been carried out. The cylinder is internally pressurized and is under three types of load ings, namely, axial compression, bending, and transverse shear load. The stiffeners are not very close but are discretely located and the eccentricity of stiffeners is disregarded. The method of the solution is carried out by the use of nonlinear large deflection theory and the effects of initial imperfections in the straindisplace ment equations are considered. The Ritzmethod is used xi to find the governing equations of instability of stiffened shells subject to the three types of loading. Numerical examples for cylinders with inclined stiffeners at different angles are worked out. The minimum strength of shell versus different rigidities of stiffener are also calculated. The relation between the minimum strength of pressurized shell and internal pressure is plotted. The computation was carried out on the IBM 709. Some tests have been made on the shell with inclined stiffeners at angles of Y = 300, 450, and 600 under axial compression and bending loads. Some of the experimental data are plotted to compare with the analytical results. $ xii CHAPTER I INTRODUCTION Interest in shell buckling phenomena dates back to 1858 when Fairbairn (1)* performed his experiments on the buckling of cylinders under uniform external pressure. In 1888, the first theoretical treatment of the problem was published by Bryan (2) and Love (3), and since that time, many investigators have been attracted to the various prob lems of thin shell instability. The equations for thin shells were discussed by Timoshenko (4) in his wellknown book. The study of stability of isotropic cylindrical shells was advanced by Southwell (5). In 1933, a set of equilibrium equations for cylindrical shells subject to torsion was derived by Donnell (6) based on the theory of small deflections. A simplified method of elasticstabili ty analysis for thin cylindrical shells was also discussed by Batdrof (2). Underlined numbers in parentheses refer to entries in the references. Then Suer and Harris (g) applied Donnell's eighth order linear equilibrium equation to the problem of the stability of cylinders under combined torsion and hydro static pressure. They obtained a solution with Galerkin's method based on an assumed radial deflection function in the form of an infinite trigonometric series. In 1961, Seide (9) had presented an analysis of the buckling of cylindrical shells subject to pure bend ing. However, these classical analyses are valid only for infinitesimal deflection. When the deflection ip not very small and has a magnitude of wallthickness, large deflection theory must be used, and the high order terms of deflection are to be considered. In general, small deflection theory predicts buckling stresses approximate ly three times as great as those found by experiments. This evidence is particularly obvious for cylindrical shells under axial compression. The discrepancy between the small deflection and experimental results can be explained by use of large deformation theory which was advanced by Donnell (10) and von Karman and Tsien (11). In Ref. (10), Donnell also considered the initial imper fection in the shell to cause lower buckling load. A diamondshaped buckling pattern represented by a function of arbitrary parameters was first proposed by von Karman and Tsien (11) for cylindrical shell under axial compression. Their results agreed with the experimental results. The same problem was also solved by Kempner (12) who used the Ritzmethod, but improved the von Karman and Tsien's solution by considering five free parameters. An investigation of the buckling of cylindrical shells by use of Galerkin's method was also studied by Ekstrom (13) and Lu and Nash (14). The general case of the axially compress ed orthotropic and isotropic cylinder with internal pressure has been discussed by Thielemann (1). The general equa tions in tensor notation for isotropic shells were derived by Dill (16) who also included the effect of initial imper fections. The effect of initial imperfection on the buck ling of cylinders under different loadings was studied in Ref. (1i), (18), (19), (20), (21) and (22). The general instability of ringstiffened or stringerstiffened cylindrical shell by the use of small deflection relations have been investigated by many authors, Nash (23), Alfutov (24), McKenzie (25) and Kan and Lipovskiy (26). Further study in general instability of ringstiffened cylindrical shells subject to external hy drostatic pressure with a comparison of theory and experi ment was discussed by Galletly, Slankard and Wenk (27). A detailed treatment of the buckling of stiffened shells has been discussed by Hedgepeth and Hall (28). The problem of stability of orthotropic shells was studied by Becker and Gerard (29). Further study in general instability of orthogonally stiffened plates or cylindrical shells has been discussed by Huffington, Jr. (30), Neut (31) and Becker (32). In Ref. (32), an analysis of orthotropic cylinders was made by Taylor whose derivations were based upon the use of a form of Donnell's equation. Recently, an analy tical investigation on the instability modes of ortho tropic cylinders subject to compressive or bending load was made by Block (33). He used the Dirac delta function for rings in the linear equilibrium equation and considered the pressure load. The large deflection relations applied to the prob lem of the instability of ringstiffened cylindrical shells derived energy method was investigated by Lee (34). 'Experimental work on the ringstiffened cylinders or unstiffened cylinders under different loadings has been performed by MaCoy (35), Peterson and Dow (3), Lundquist (3.), Harris (38), Fung (39), and Goree (40). The theory of anisotropic elasticity has been dis cussed by Hearmon (41) and Lekhnitsky (42). A general discussion on the theory of anisotropic shells has been made by Ambartsumyan (43). A linear solution of aniso tropic cylinders was investigated by Cheng and Ho (4). The purpose of this study is to investigate the general instability of cylindrical shells with a system of stiffeners which is mounted at various angles. The cylinder is internally pressurized and under one of the following types of loading: a. Uniform axial compression b. Uniform bending c. Uniform transverse load. The stiffeners are not very close but discretely located and thus the anisotropic relation does not apply. The Kirchnoff hypothesis is assumed; the eccentricity of stiffeners disregarded. The method of the solution is first to assume the deflection function, then the stress function is found from compatibility equation which is carried out by the use of nonlinear large deflection theory. The expressions for the energy stored in the shell and stiffeners as well as the work due to the external loads are obtained. The large deflection terms and the effect of initial imperfections in the strain displacement equations are considered. The Ritzmethod is used to find the governing equations of instability of stiffened shells subject to these three types of loading used. Hence, the minimum loads are found from these equations. Numerical examples are worked out for the cylinders with the inclined stiffeners at different angles. The mini mum strength of shell versus different rigidities of stiffen er is also calculated. The relation between the minimum strength of pressurized shell and internal pressure is plotted in Fig. 15. The computation was carried out on the IBM 709. The general computer programs written in Fortran II to find these minimum stresses for all three types of loading are listed in Appendix B. Some tests have been made on the shell with inclined stiffeners at angles of Y = 30, 45, and 600 under axial compression and bend ing load. Some of the experimental data are plotted in Fig. 18 to compare with the theoretical results. CHAPTER II BASIC RELATIONS 2.1 Compatibility Equations In the present investigation, the following assumptions are made: a. Elements normal to the unstrained middle sur face of the shell remain normal to the strained middle surface. b. The material follows Hooke's Law. Let x and y be measured in the axial and the circum ferential direction in the median surface of the underformed cylindrical shell (Fig. 1), w equal the total radial deflec tion, and wo represent the initial deflection in the radial direction. The straindisplacement relations are e 1 +_ _.___ x x 2 2 ( ((1) (: i ( f 0 a 2y 2 2 B R ex bJ Lk Lr bur r D J. bura Z ~ ^ h rx ax b xc ar In the above relations, the higher order terms of deriva tives involving u and v are neglected, since these dis placements are small compared to the radial displacement w. The stresses and the strains in the median surface of the shell in the case of plane stress are related to each other by the following equations: E 5xET^ (Et6x) 6Y 1 (2 +,) ) E _rE2' ( ) (2) x4 2(1+)) By substituting Eqs. (1) into Eqs. (2), the median surface stressdisplacement relations are obtained E= C 2 CAX 22F I r 22, : 2^ ) + , + aLk, 1 ,iU r I 7W+ ) (xW +jr ((AT. ZX$= ax 2ar x 3 (3) The conditions of equilibrium can be satisfied by using the wellknown Airy stress function F which is defined as S__F_ 7'" F X=E E I 1 6*r F (4) Eliminating the variables u and v in Eqs. (3) and (4), the following relation between stress function F and the radial component of the displacement w is obtained. ( + F E "co x+2  y 2 ax (5) This equation expresses the condition of compatibility be tween stress and strain. It was first obtained in the pre sent form by Donnell (10). In general, wo is unknown. For simplicity, w is assumed to be proportional to wo Ref. (10), (17), (20), and (22) ., Thus one can define OX = imperfection ratio (6) where r is independent of x and y. With the relation from Eqs. (5), (6), the compatibility equation is expressed ass 41 b 2 eLa r ? l T fz  2.2 Enerqy Expressions The strain energy in the shell and stiffeners and the potentials due to external load are found below. a. The extensional strain energy in the shell can be written as: x1 "^ F JJ F 2 F] dxdy b. The bending strain energy has the following expression: U DL27rR Z247+ 2 a b 2Jo [ o xX X2 aX 2 2 ?(rx. aJ),u( a) __ ]9 (9) where E t3 D= E(3 12C11 ) c. The potential due to edge bending of the shell +2((1+2 rM L ZIM Cos ]1.u ]&,Ct f o b COSc (10) where Ub is applied peak bending stress and ax~E \ b'F b+F\ Llb1 (,.rx, d. The work of the external force applied at the ends of the shell can be calculated as the product of the applied force and the change in length of the shell as follow: TT = (Uf~fl F ( 3AF\jU (11) tJoo 1 ET Z, 2 W F 7 where Cc is constant stress over the shell thickness. e. The potential due to the internal pressure p is S, 1< 1 (., a) dxda Uo ( (12) The change in volume due to the displacements u, v are neglected (11). f. The potential due to the edge bending on fixed end by transverse shear load P which applied at the free end (Fig. 3) is  L 1o rP CCS ) d =l x COS )2 ITdfol X2R Ex 2 'x (13) g. The potential due to transverse shear load P applying on free end is STT oJjo 1 E 3xa x ,, where P is average transverse shear load. h. The bending strain energy of stiffeners. The stiffeners are assumed in parallel with the yl, y2 coordi nates lines and the principal directions of the cylindrical shell coincide with x, y lines (Fig. 2). Let the subscript k be used for kth stiffener which is inclined at an angle 1I with the generator of the cylinders, and is parallel with yl line and normal to y'2 line. Thus the bending strain energy in the kth stiffener is (15) where Nk denotes the number of the stiffeners in (1  direction, and Ek Ik, represent Young's modulus and the moment of inertia of the kth stiffener, respectively. The eccentricity due to stiffeners is neglected. The limit Ik is the length of the stiffener in Yl direction. Similarly, the bending strain energy in the jth stiffener which is parallel with y2 line and normal to y'1 line (Fig. 2) is j=1 Jo = (16) The subscript j is used for jth stiffener which is inclined at an angle of '2 with the generator of the cylinders. Where Nj denotes the number of the stiffeners in I 2  direction, Ej, IP, represent Young's modulus and the moment of inertia of the jth stiffener, respectively. The limit Ij is the length of the stiffener in 2 direction. i. The torsion strain energy of the kth and jth stiffeners are UT, = 2 J,, \y 2=0 (17) _= 2 {J II Ur I;Id I (18) where G and J represent the shear modulus and the polar moment of inertia of stiffeners, respectively. The sub scripts j is for stiffeners in 7 2 direction and k in Y1 direction. In the present analysis, the inclined angles, Y1 and 72' are considered in axial symmetry (i.e. = = ) CHAPTER III INSTABILITY OF A STIFFENED CYLINDRICAL SHELL UNDER AXIAL COMPRESSION AND INTERNAL PRESSURE 3.1 Deflection Pattern and Stress Function Let a deflection shape assume the form (12)s W = b,+ b2 COS C COS' +b+COs 2M t+ COS y (19) T + + R where m and n are the numbers of waves in the axial and circumferential directions, respectively. In Eq. (19), bl is not an independent parameter but is used to satisfy the condition of periodicity of circumferential displace ment (11). It is found later to be a function of b2, b3 and b4. The boundary conditions of the displacements at the ends of the stiffened cylinder are disregarded. This simplification is justified by the experimental findings of N. Nojima and S. Kanemitsu as discussed in Refs. (11) and (12). It was found that there is no appreciable length effect when the length of the cylindrical shell is greater than 1.5 times the radius of the shell. Therefore, the long stiffened cylindrical shell is considered hereon. The corresponding stress function is _2 pR j~ = Be 2 1 X Cos 32 cos + oCnocos St ,3c Cos M +x o Co rtl 3rC C0 O5 The coefficients all, a22, a20' a02, a31 and a13 are found in term of b2, b3 and b4 from the compatibility equation (22). These coefficients are found to be: S ( r) (  aonEl 22 ctoz= E*I F')32,u2 2) b 2 a =. = (i 2)2)(b+ ) ._ E 2 (ci+^( +,'UZ) b2 b4  31 2,, .z b rF2 2,4  (1+4,2 b3b R mZ  (21) A ==2,,3, 4 (22) a cos cos os a, 1os cos + a22.COS cos (20) a, =  where b, b, ~t 'It Physical considerations lead to the conclusion that the circumferential displacement, v, must be a periodic function of y. This is expressed in the condition that v/ by contains no constant terms or functions of x alone. Using the second of Eq. (1) together with Hooke's law, the following expression for ?v/ 3y can be obtained: __ 1 i6 )f6 1(pA, /^1 /,\ (U(j (23) By substituting Eqs. (4), (19), and (20) into Eq. (23) it is found that E 18 R 4 R I oR (24) + terms involving periodic functions Since v must be a periodic function of y, the constant term on the right side of Eq. (24) must be equal to zero. Thus, b, (25) where R  6 (26) 3.2 Expressions of Total Potential Energy After substituting the expressions for w (19) and F (20) into the relations given in Chapter II, section 2.2, the various energy terms are obtained after integra tion. They are UeR 2 1+/Z z 4 (1/) S + f+ 2,d U + aI, + Ctas TrEPL O )  g+ (ba C + )+ (9+4M C+(1+5) 4 (27) UbR + IA + 16B ; +;Z fTE aL 24(1) 2 3 4 (28) (29) Pz^( ^) + ?(( T +6e) (30) =Eix i2 r b((ce C, + R 6, C) +32 32 (;; ; + 1 +R, (31) S(1 r SP _ (F j'Z + (Ca,uA C)2 (C4 f + 4C3c3 ( 6,4f b2) (34) where E ^ E~Gk r ^G=i E E E = E Ei JE  IJT  Ij 1 jT*  T1=7^ ST F , J ~ fR tr E ..L :TEt3L 2 bz (ct + c + 6 4cic) S)] +R2 (ij r r ,=1 JT E*3L (32) Jr EPL (33) 3t2 42 + 32 (,,4i: + b G6 q[(+C+ACi+ cc(C,c,)z +(c, Ac .1(c(+a c + 66cci(4+ fC)+ R, 16 Lc' c E~~LI[ }+R, l =C =7 Ci=Cos Ca = SiN C=.3i = C3=Cosa5, C4= SIN , S (35) The expressions R1, R2, R3, and R4 are very lengthy and can be neglected if the stiffeners are located uniformly and not far apart. They are listed and discussed in Appendix A. The total potential of the system is the sum of the strain energies and the potential of the applied loads, as follows: U, = Ue+T+ T+ +Tjt ,*+t.iT, ,,, In nondimensional form: r +R 2 u( ++Z) JT E'L 4 8 Z 4(if~+' )2 , a 4 + 4 + )+ + '(1+q, eY(1 a is] 9 (l++ 16 ,U4;' +b!')] +R, 2 c?+ 6 Uz2C' CZ + U1 4( + + + c C 4+ n 4y (c+ c) (c c ) 44(c f C44+ c,) j=1 + G4 J c(,+(C^ t ')2 C i+ (C, c(c2 C(C ] z 3.3 Minimization of Total Potential Energy to each of the arbitrary parameters must vanish for equi librium, this yields 4 o = AO 0 44 z (2 &bjb16} (C. +A^(4 a +(C3 C4CI(C4 +,u c3 + 4 C A++ 624 +R,+ ? Rzt R3+FR4 (37) 3.3 Minimization of Total Potential Energy The variation of the total potential with respect to each of the arbitrary parameters must vanish for equi librium, this yields (38) After differentiation, the following three equations are established: + i Ef(Cf+ 6COCc +,4c 4E + 7 l ( e + 6_Ae C31 C* + C.le) It 4 T  + C [2 (24r)(ZT3+b4) + 4al b + i (i+1 6L (1 +A)2? 2 a a(i+X2 b( (i+ ) (+r) 4 4 (f+r) b+ _ + + p ( L a+ 4 +A1+ y ( 39) z  (iF) 4(11 ) b3 + lU4 + c4 + A A4 (Iff r) 2 (i1+A?)T 2 (i +x : +1 i+ W 2 4 2.A ( 1I+) S(I gsy ^ ~Wk c j i * (40) _6 l 4(+ bA. 1r b. 4(1+A'7 g A4(1+r) 2(1 +A9Z 2 6 Ir 1F) i *tM" 2 2Al~itr) f (i $U2)2 1,~, + 14r) , 4 4 S29+3( 3 + z 2(9+.Y  4N Nj + t EI c), + j= 1 1 3 + 4 cT rc2r c*"i i^c ETjja c, kk4 c, C For brevity, Eqs. (39), (40) and (41) may be rewritten as: , =Ar t(A,+A,[+A4,)+, A5 Tr =b ++ ( +) ' =H,+ (H2+ H4) where = 6o,,2" A, Az 12(1 'e)Z (1I+ )2 A3 ([ (2+)(2 1' ) (1 +*)2 (41) (42) (43) (44) N + > lk= +2 J + 2 (Hs + ) + i 1a 1 zi^4 ^ + j E i3 (C, + 6uc:c +,. cl) jt 2 I E I*,, [ (C,+Ca,) (Cz,c,Jt. E l4 +x (C4+ ) c4(c 3)c)f +iAcf] (cs,ac4f(cC++,c,)] (4 o 9 /(42 A4 =B, (L= = (i +i + S3 2 4 4(i+e)2 (1 + X) (i~f 72 2 Ce ai;E,+4cA C c,2 cU + "U4 1Tj0 cy* , c1j C3 C 4iz jz=l (46) A4=4(i + "f) I (+,) 4 + AJ) As= (1+ r)l 16 (C,A cC (c + k=l H, 3 (1z) 1 Ha 4 H4=(1+r) + 6 la+ 8 H3 + + G %C C, c + (47) Ejjl 1 c C4 E7,I c^ %=^3 (48) Eliminating b2 and from Eqs. (42), (43), (44), the following equation is obtained: M,I T+ M$ + M3 =0 where Az 4X, c4 03 (49) Ej I)i 31~ (3 (1+ U  A5) A s) F2 D3 /RD M, T f 2 a fD  (r F + 2F l(5+B A5) F3 t F( x (H! +5) A2 As (B,+ + H )  P1 (i__ pl pz F32 IFD3 + Dj (H+ + + A _ ZPaFD 12 + z+D (50) As 34A5? + F3 :B,4 H, + F. (55+ "? B61 As) / *) 2 (B + t) X 2 = A, ( Bs+ B3c) (15+ (51) SM, (H,+ (, 86\_ B1 7t/ Ps3= (Az + A3 + \A4V Ht F2 = A (Hs ) H, A F, =(A2 + A.3 + A41) (H, ) (H + H4Lz) A5 (52) Equation (49) is the governing equation for the ' stationary load of a stiffened shell under axial compression. The numerical calculations have been carried out on the IBM 709 and the computer program used is attached as Appendix B. CHAPTER IV INSTABILITY OF A STIFFENED CYLINDRICAL SHELL UNDER BENDING AND INTERNAL PRESSURE 4.1 Deflection Pattern and Approximate Stress Function When a cylindrical shell is subjected to either pure bending or eccentrically applied compression, an approximate form of the deflection pattern is assumed Sn x 2M rr2 714 L=b,= os 6i0 (bcos Cos ,Cos b cos R (53) This approximate deflection pattern is obtained by multiplying the pattern of the axial compression by cos (y/2R) which signifies the localized buckling on one side of the stiffened shells. The effects of the end restraints are also neglected. The corresponding stress function F is proposed S29 F _6 Y + 6b Iecos + E k t+ cost cos + a COS 2 X COS2 4 zoCS2 CO52rS +a, COD + R z Cos Cos Cos +a3,coS 3 co5 I + ,13 coS5 cos3 R R (54) The stresses 6 and 6b are the average axial compression and peak bending stress, respectively, and are positive for compression. The coefficients a20, a02, all' a22, a31 and a13 in Eq. (54) can be expressed in terms of b2, b3 and b4 by the Galerkin method. This method establishes the following set of equations: IL 2JR oo O JL ZJTR Q Jo L 12~R 0 0, Loo 27Rl COs MCOS^ 0 dx o COS 2 x COS2n9_0_ 9 ax = 0 R R COS mx dC9 x= 0 R COS2^ ?j dx = 0 'mx 3__ COS Cos o L~cx = 0 R R CoS c 1x CO5 r d Oc = 0 R F (55) where S i o E [ ( :xa S 'b2 Yur r' JL + ?W *^2xa LJ LJ _b R x2 R Z~Ob It After the expressions for Lr in Eq. (53) and F in Eq. (54) are substituted in Eq. (55) and the integration carried out, the following relations are founds 0 E20 E E" o2 E, = a26 (I _  =..( ( + ) z ' E( f + AO ) " k  a' ==Ef 1 E* where E Z where r,/ , (3+) b4 (1. r2) 4 (7+ a) 3 A4 +T4+  (14 F) b b.3 4 (9qt + i (57) and bi are same as given in Eqs. (6) and (22), respectively. In order to find the bI as a function of b2, b3 and b4, we derive as in section 3.1 the relation: 3LJ0 ..b Yf (56) T ;+ F2r)] b, r f [ +(2 2 i = ~+ ) b r 1 40 + 16 ,(iF)= (+1 )+ p I 4 ( .+;) S(2d+ 1 + 4 (58) 4.2 Expressions of Total Potential Energy The total potential of the system is the sum of the strain energies and the potential of the applied loads. By adding the Eqs. (8), (9), (10), (11), (12), (15), (16), (17) and (18), the total potential may be expressed as follows: Z= e+ Ub+Tc+U,+U+ ,+UT+ (59) If w, F and b1 as given by Eqs. (53), (54) and (58) respectively are substituted in Eq. (59) and the integration carried out, the total potential is obtained in the follow ing form: JZ 2  (F Tz R 2 2 b 3(1 2 )TELk 2 L 32 r)  z 3(1F2) 6 b AI 4 4 0 16 2 1 tb r + (z .L1 c4 + 2 +2 ( [48  2 2 16 = P 16 b2 4=, 1 6  +n ) 6,4,4.Zm t  Cn2 + "n I (,+)+ + 4 3 (c1, 6,2 C IC + A C4) 2 b2 37! C, + 24 z c ~ZL 48$4+24 ,z * Ej {16 p 2Z mZ t 7n4) .3 C31 C4r + 24 McC' ci + N +2 4 (C \?*4 + 2.c ++4t 7?yy Sc  _ [19 c4j T[c, ac) (c( XAcf + (c, cf (c+ Ar [(c +'cif C 4(iMe)C2c + (ci+c)Cj+7j 32 m4) c ci + 32 2o8  2 a1 i T MZ '1 4 02 4 a +2 ,(~++ a+ + + +] 4 +6 ++ (2 + 4 m2 92 2mz  + P 2 + 224 (I +fl"^ +,a+ c:) + 4 ad+ 216 + 16,bg +] wt+ +cz1 + In4. lb :' L C c z+ C +4 + +b (48,e+ 24+) +~ 5324) c4.c X(C4*c.#)+ (C3xC+)#(C4+,aC.3)Y ]+ [ (Cz+.C4L (60) where R, R2' R3' R4 are also the portion of the energy of C4 4 the stiffeners which have the similar expressions as RI, R2, R3 and R4. Since they are neglected in the numerical calculation, their detailed expressions will not be elaborated here. From the discussion in Ref. (14) it is indicated that m has a greater magnitude when R/t is greater. In the present analysis, we are concerned only with extremely thin shells; hence this ratio is large. Therefore, for practical purposes 1/m2 and 1/m4 are negligible compared to unity. Thus, from Eq. (60), the total potential can be simplified as follows: = 2 2 2 f 2 b = ' 6 6b 236 (1r) U? 2 36 32 2r (4  ( Ir1) Az 2 (1+ )2 +ai, 4 4 az o 12zo0 .a +t aoz S ( q + 2 %+ 03 + C,4 + + C,'cc) +24+p t+ct+ 244 +c] +A*4ct+ 6 c4) j I6C~ xt c [( ,+uc,fc(c,a,f +c(C, , C.t2 (C c]+24 W. 1 2 ' + c { (c., J, C4 (C4A c) + U b 41 +1 6= iP2 L B + (c, A c4 (C4 +,U C3] + 24 ,3 AM cfC4 +24,C. c + aa 71 ) 3 +~~;)I ib (t.+,uzJ +4 (,"A ) t iz "l ,) where 2 1 a^=0 lb 3 A2Z a 16 [f; +2z4; +*c 24 ;] I6w*^x ^l^ (61) I 6 zg r2} a 4 (1C+2 a22 p + Z ( 16 i( +cr 2 3 a0 (I r) 3(b a, ( i+ 3.4 T 8 2(1+Aa )2 r 3 3 AA 2 b3z b 3 AU a13= ( (fff)) b b 4(1+ J)A)2 (62) 4.3 Minimization of Total Potential Energy The total potential energy must be a minimum when the structure is in equilibrium and this condition leads to the following equations o 0o o S63 b 2) b4 (63) Substitution of Eq. (61) into Eq. (63) leads to three equilibrium conditions as follows: 1r A, + (A + A3 +AA32) + b As (64) S2z o"= +, + 8B41 + b6z =T  where * =I + 23+  (I (1+^ )2  A  A2 f+ ) ~I (1 +)2 A4=(+r) I+q( {4 ( L.+U zl)l + ` 256 S = 2 El [( 4 CN a^ccr IcI4C) N+ EI [13 (C4+6 6'azc;c +eC4)] t.^Z GfsCC, +AAC^jGC.AtCcfjc,tcc,+tcj+^ + =r 8 =6i~ t8[c~~c(c)t~,p~G~C) (67) (65) (66) 7) t~Jl it I+) + i T 4 + a r/ + + 2H+ 4z+ b, (H+ + 3 8 3 = ' 2  = (F+) A+ s. 5)2(1321W ?2 S16(1+ A ( )2 + =^ i^ '4 + Zcecj~ )+ 32 (1 +AL' ) a N 44 3G j ,OC32C4 T = + 33 2(1 H, n 36 A H4 32(1+ + H1= (1+i r) 6 128 ), + )2 a1+) 3AA2 16 (ltAjl (68) (q+,te)a1 32 S = 3 E6JI.& + 3 TA N' C2cf + 3EIi c cc l + 3 Eliminating Sand 12 from Eqs. (64), (65) and (66), the following equation is obtained: MI3 =0O where As) Z D D, H M2' a2 )P +( + 2(15+ Fz D3 D z2 C,35 + $ 5As + ) F1ta AsBs + E *+y ^Z H,t ( RS (69) (70) P, D3 F3 ]6 _ 7;1 i M 2 Di(B,+ t)] F  8,+B +i~ 54 H )(H + ( + AS)2 2 = = ( + ,H +; + ,+ A 52 + 53 2 Az (71) =B zt )H,A F3 7 ( 2 F3= (A+A31+M A )(S+ ()(H+H4 A, (72) and = 6 (73) Eq. (70) is the equation for the stationary load of a stiffened shell under bending load. Numerical calcula tions of the minimum stress parameter <2 are given in Chapter VI. CHAPTER V INSTABILITY OF A STIFFENED CYLINDRICAL SHELL UNDER TRANSVERSE LOAD AND INTERNAL PRESSURE 5.1 Deflection Pattern and Approximate Stress Function When a long cylindrical shell is subjected to trans verse shear load at one end, an approximate form of the deflection pattern can be assumed as in the bending case in the previous chapter in Eq. (53). The deflection shape is = b, + cos2 ()(b)bacoscos +C coS +b45c Corresponding to this deflection pattern, the stress function F is proposed =o + + a,, COs 7L + 6t2z COS 27n COS 29 + 2 COS 2 .a o5n a2z Cos2xCoss + +a0 Ccos  (3 Co5 3M COS^ + ,3C05 COS (74) The relations between the coefficients all, a22, a02, a20, al3, a31, and b2, b3, b4 are given in Eq. (57), and BE is the same as given in Eq. (58). 5.2 Expressions of Total Potential Energy The total potential energy of this system is the sum of the strain energy and the potential of the applied loads. = UtUIb+UT +Ui+UT,3t U J (75) If w, F, and I as given by Eqs. (53), (58), and (74), respectively, are substituted in Eq. (75), and the integration carried out, the total potential is obtained in the following forms T, R 2 2 ( 2 2+ T3 ETAT L + 5 '4 2(1i 3 I (+ P 2 22 b6 P P 41 62, b 16 4 6 4 4 2 2 2 2 (1+a z 4( 1+ f 2 g ZZa + a,, 42 + a22 + a20 + ioz + *+ l4f +2 ^ ^ ^ 6" B^ a2 +U 2 (I+ 2 + (I r )2 ) S 4 p2 13 4 g 192 (I) + I2 + b(44 24 + )+(76) (76) where .J LU .'U ,lTUTTj obtained in the bending case Eq. where and l are given in and PL E JT Rf For the chapter, 1/m2 in Eq. (76). trained in the are the same as those (60). Eqs. (22) and (6), (77) same reason as mentioned in the previous and 1/m4 can be neglected compared to unity The total potential energy is finally ob following forms i5 2 2 2 + 1) 22 ,3 P p (1i+9) P P + T+ ( P2 2 P2 402 K?^ ^^. ^) ^cr 4(t^") +/ S^4C^+^)+ it 4 2 192 (1 .0 24] (78) + U.+ +T, + U where all, a a02, a20, a31, and a13 are given in Eq. (62). 5.3 Minimization of Total Potential Energy By variation of the total potential energy in Eq. (78) with respect to each of the arbitrary parameters, three simultaneous equations are obtained. From these three equations, two arbitrary parameters (b2 and ) can be eliminated. Finally, a governing equation for.the stationary load of a stiffened shell under transverse shear load is obtained: 2 M1 M, 23+ M3 = 0 (79) where P (8 D3 ? (80) MI, M2, and M3 are given in Eq. (71). CHAPTER VI NUMERICAL EXAMPLES The minimum stresses are to be found from Eqs. (49), (70), or (79) for the various types of loading. In each case, the stress parameters ( f 2 and 3) are func tions of three free parameters 1~, 1, and U.. While A. is the wavelength ratio, the expression i and 1,' can be considered as deflection parameters ratio. 'The minimum values are found numerically at various inclined angles, stiffeners rigidity and the internal pressure. Numerical calculations were made on the IBM 709. Program I is for finding minimum stress ac for axial compression case. Program II is for finding minimum stress parameters, 42 and 13. The minimum stress can be found with any given numerical data. In the following, some examples are given. 6.1 Cylinder under Bending and Internal Pressure or Transverse Load and Internal Pressure In this part of calculation, the cylinder is assumed perfect, i.e., J = 0 [Eq. (6)], and the angles of the inclined stiffeners with the generator are equal, i.e., (l = Y2 = Poisson's ratio is 1/3. The general characteristics are assumed: 1,= Ij I r=j T. Tj K E I N j (81) By substituting Eq. (81) into Eq. (70), the value of 12 as a function of n ~1 and ,4. is found. First, 2 versus n is plotted for various values of UA. at a fixed value of l1, in Figs. 4, 5, 6, 7, 8, and 9. Minimum 12 found from each of these curves is called 12' n It should be equivalent to the value found from the relation S2/ b = 0. Then (2, t versus / is plotted for various Y 1 in Fig. 10. Minimum 2, found from each of these curves is called 2, ,A, plotted in Fig. 11. It should be equivalent to the value found from the relation S2, /)a =7 0. This minimum value is the dimensionless critical stress ( b + 3/2 6 )cr for the shell with in clined stiffeners at an angle of f = 450 with the generator. By the same procedure as in the previous case, the curves of 2,I versus t 1 for various inclined stiffeners at an angle of Y = 00 (stringers), 300, 600, and 90 (rings) are found and shown in Fig. 12. The minimum value of each of these curves is plotted in Fig. 13. The relation between minimum stress parameter 2,. and stiffeners rigidity, E I A is shown in Fig. 14. The relation between minimum stress parameter 2, and internal pressure p is also plotted in Fig. 15. If ;3 is replaced by 2 in Eq. (79), the equation is the same as the one obtained in the bending case. Therefore, the minimum stress for the case of transverse shear can be found by using 3 instead of 2, , in all the figures from Fig. 4 to Fig. 15. Where 3, 2.)cr. 6.2 Cylinder under Axial Compression and Internal Pressure By the same procedure as in the previous section we can find the minimum stress, ( from Eq. (49) by Program I in Appendix B. In this part of calculation, a numerical 47 example is given for comparison of the effect of the imperfection in shells at various inclined angles. This is plotted in Fig. 16. CHAPTER VII EXPERIMENTAL INVESTIGATION A group of tests for the general instability of, rings and/or stringers stiffened shells have been made by many authors, but up to the present time, no work has been done on the experimental investigation of the general instability of inclined stiffened cylinders subject to axial compression or bending. Therefore, it was necessary to conduct a series of tests in order to compare experi mental with the theoretical results. 7.1 Models The models used in these series of tests were stiffened cylindrical shells constructed from Du Pont Mylar of 1000 gage (0.01 inches), type A. Tests indicated that the Mylar sheet has a Young's modulus E varied from 550,000 psi to 780,000 psi and Poisson's ratio of 1/3. In the numerical calculation to follow, the value of E is taken to be 700,000 psi. All of the models were made by rolling the Mylar sheet around a thickwalled steel tube and joining the ends with a 3/4 inch wide strip of doublefaced Scotch tape. It appears that these joints might stiffen that part of the cylinder appreciably. However, reports from various investigators indicate that this is not the case, since buckling waves appear across the joint with no noticeable change in pattern. The present tests have confirmed this observation. The inside radius'of the cylinders was 4 inches, while the length was 9 inches. The stiffeners were made by cutting the Mylar tape to a bandwidth of 3/8 inch. The thickness of each layer of the tape was 0.01 inch. All the specimens had stiffeners made of two layers of the Mylar tape (i.e., nominal thick ness of the stiffeners = 0.02 inch). The doublefaced Scotch tape was used to join the layers of Mylar tape. It was also used to bond the stiffeners with the cylindrical shells. The adhensive tape was not effective when more than two layers of Mylar tape were used as stiffeners. The spacing between two neighboring stiffeners was 1 1/4 inches. The stiffeners were inclined at angles of ( = 300, 450, and 600 with the generator of the cylinders. In order to provide additional information for the numerical example of the previous chapter, the dimensions of the stiffened shells used in the experiments were the same as those considered in that example. 7.2 Test Result The cylinders were mounted vertically on the test machine. The upper adaptor has two pivot pins fixed diametrically at its edge. The pins are supported by horizontal bearings which are fixed to the frame of the test stand. When a moment is applied to make the upper adaptor rotate about its pivot pins, this transmits the bending moment to the cylinder. The lower adaptor is connected to a circular plate which can be moved up and down so that the axial loading can be transmitted to the cylinder. Bending load is applied by means of a lead screw which pulls a cable up through a bearing which is fixed on the frame of the machine. The bending moment and the axial compressive loading were found after calibrating the readings from a strain gage indicator. The test results of the stiffened shells subject to axial compression or bending without internal pressure at an angle of dI = 300, 450, or 600 are tabu lated in Table I. Some of the results are plotted in Fig. 18. The deformed pattern after buckling at p = 0 are shown in Fig. 17. The buckle pattern formed across the seam with no distortion in shape and none of the seams failed during buckling. CHAPTER VIII DISCUSSIONS AND CONCLUSIONS In the previous chapters, nonlinear analysis on cylinders with inclined stiffeners have been made by the energy method. From the numerical examples the following observations are made: 1. From Fig. 13, it can be observed that the mini mum stress parameter 2 increases with increasing stiffeners' rigidity, and the optimum inclined angle varies with the rigidity. In the present example, it has been found that the most effective inclined angle is in the neighborhood of f = 600. 2. In Fig. 14, strength of cylinder with stiffener inclined at Y = 450 is compared with that of ring stiffened cylinder. At smaller stiffeners' rigidity, the ringstiffened cylinder is stronger, but at higher rigidi ty, the inclined stiffened cylinders have more strength. The minimum stress of ringstiffened shells approaches to the buckling stress found from smalldeflection solutions as a limit, while the inclined stiffened cylinder continues to increase with increasing rigidity (E I ). In the same figure, the consideration of torsional rigidity (for example K = 1/2) is compared with the result including the bending rigidity only (i.e., K = 0). The difference between these results increases with rigidity, E I t However, the general relation of the minimum stress with other parameters will remain the same. 3. Example of the variation of 2 versus is shown in Fig. 15. For other combinations of Y, K and E I T, the curve will be similar but different numerically. 4. The minimum stress of cylinder under axial compression versus the various inclined angles is shown in Fig. 16. Imperfect cylinder (r = 0.3) has lower strength as expected. The evaluating of f can be referred to Ref. (22). The effect of imperfection for stiffened cylinders is smaller in comparison with the unstiffened cylinder. 5. The buckling patterns of the unpressurized stiffened shells subject to axial compression are shown in Fig. 17. The deformed pattern after buckling is diamondshaped and across the inclined stiffeners as expected. 6. In Fig. 18, it can be observed that the theory for the bending case is in reasonable agreement with tests on the Mylar inclinedstiffened cylindrical shells. This study has presented the approximate solutions by energy method. Nevertheless, the results of this analysis should give some insight into the problems of the general instability of stiffened shells with inclined stiffeners. TABLE I TESTS OF STIFFENED THIN MYLAR CYLINDERS UNDER AXIAL COMPRESSION OR BENDING WITHOUT INTERNAL PRESSURE Cylinder Radius = 4 inches Mylar Nominal Thickness of Shell, t = 0.0075 inches Thickness of Stiffeners, t1 = 0.02 inches Inclined Dimensionless Dimensionless Dimensionless angles rigidity of axial bending stiffeners compression stress_ i Ti lc 6b 450 0.059 0.267 0.306 0.274 0.317 0.117 0.282 0.312 0.294 0.329 0.308 0.317 0.302 0.306 0.47 0.287 0.321 0.339 0.382 0.322 0.323 0.94 0.368 0.435 0.328 0.388 0.335 0.412 0.336 0.376 1.47 0.333 0.365 0.335 0.365 ___0.339 0.358 300 0.124 0.281 0.306 0.315 0.321 0.992 0.302 0.353 0.322 0.335 0.314 0.353 600 0.134 0.288 0.388 0.322 0.322 1.072 0.302 0.353 0.322 0.370 0.322 0.370 Fig. 1. Coordinates and Displacement Components of a Point on the MiddleSurface of the Shell. x yl Y2 _y Fig. 2. The Coordinate System of the Stiffened Shells and Stiffeners. (a) Under Axial Compression P c ) (b) Under Pure Bending M (c) Under Transverse Shear Load P Fig. 3. The Dimensions of the Stiffened Shells under Different Loadings. 0 4J u 1 II \ + 0* ^ S\ .IS O 0< 0 14 0 II 41 el' 00 S\o Co N 41H S\ H Ic W r O0 4. 0 tp I c a ok 0 * H 10 ^^ ^^^ ^ ^ ?1 "" ^ ^ ^^ ^ ^ 1 9 N ^'^^'" ^^^l a ^ nl ^^< r9" V d d d 0 *J 4i 44 0 0 *4 4J 0 0f S S, 0 b4 II 0 l' 0 II 14 II to II I1 '1 >0;= I p "i i , N O a S ; 0 0 H 41 c rI 0 0 r. )H 0 14 0 m3 M, 0 II O 0 L II IN 0 I ' u, v N RH __ __ __ U, C; 0 F3 0 0 II 04 I4 C14 0 0 r; II 44 3j 0 o 0 k 4IJ CO II 0 0 00 0 o 30 w *I 0 4J 0 ? II u to Cel 62 In eq 44 0 N 4 1f 0 \ 0 II v.I 0 4 1. ra \ i o 0 L eq o H ooo cc '. in *r H  B B B cc 0 0 0 0 0 63 in N co 0 Ila, 0" 1" 0 S II 0 S1 I SIto Cl 4 I I ri U.N 40 NN N S 0 1 O CO 0 0 0 co r4 0 0 0 II II U II II II II i co r. .. q S 0 0 rD in * o o e1 N S * 0 0 Sl S o 0 0 o H D. 0 S4J &I oil 1 I 4 1 1 0 k to 0 0 Vc 14t 'B 41 HQ I I I I I I I I I J i I ! 65 ,N r  4J U I O II 4 00 ^ IPI o0 0 ^ as  O O t N N qo 1* +1  r0 HI Ak ' * C4 N U 44 N 44 0 co 0 0 I4 f4  * : C4 i 4 0 0 o N o 4 o N 0 W 00 Se 1H S e1, I V 4 0 / 14 4. 41 N * 0 0 .0 M 0 HH g ;m V 67 NI 0) II jH 0 li .0 4) In a a S\ 0 r 04 0 o 0 o0 # 4J r i M o 0 .0 0 0 \o *4) o#a So cn \D r m r 8 o a C; o h( 0 o I \I \ I 0U In i0 1t 4J 41 a0' ri 4 I W o M aO LA m C 4 * '1 00 0 0 C0 0 0 0 (\ N 3 W4 0 $ II o n4 I l ol  I 4'4t) II o :ci elqa to II $4 So r m0 a 0IH S00 0 Ht 0 Ln 4o Ln o C; 0 IoL C.' 0.4 T = 0 T'= 0.3 0.3 0.2. 0.11 I I a i 00 300 60P 90 Fig. 16. Comparison of Effect of the Imperfection Ratio on the Stiffened Shells Under Axial Compression (E I = 2 K = 0 p= 0) I I I a4 N *H o m C ) HE S0 a4 != E 0 U u *d H f( U . 4m 1i, i ~clI So r0 S I o0 < 0 < 0 m I~k r4 CO 4 9) (a k 14.C S400 4 44 S.4 Oa 1 1 0 044 4C1 U 0 tn 0 9 N r0 HM APPENDICES APPENDIX A APPENDIX A ENERGY Ri IN STIFFENERS To neglect the energy terms R1, R2, R3 and R4 in Eq. (37), we may refer to Ref. (3). In that study, the numerical results have indicated that the minimum stress is only slightly affected by these terms. Also, it is to be noted that the minimum stress is nearly independent of the locations of the stiffeners. In the present analysis and experimental results, it indicates that the shells buckle in multiple waves pattern. This implies that m2 and n2 are much greater than unity. Moreover, from the experiment the buckling stress is not affected by small change in the stiffeners' locations. Thus, the summations of sine functions of the energy of the stiffeners may be assumed very small when compared with the constant terms. The approximate solution can be much simplified by neglecting these Ri terms (i = 1, 2, 3, 4). However, the expressions for R1, R2, R3 and R4 are expressed below for the purpose of reference. *t (Flr 1 2 N {C S(c, + Cv )(c, Czt x fe~I z *' L mC X LZmCi Sil m7nc, ) + S sin Atz LZn C ( a  z a~ L (C + LCjP Ci FL^(jinC) sin(7enc A) SL(mc, +3n C.) s c 3 ) ( xf 2zb, R sm7nmclnc' 4 z A z (ca c) C L ( c ) (L(3 cn,) LmC n'C .(mq3 C. ) n L rnC+nCz) z Ta cI ( cfac~x LL(smC ) .A3in 1? si((.A) +L.R s n( nc,4n (A)ns acc L ( nc) ?. L Z ,c++n0 2) X Sin(z m c C2)()+" ZL(m,c) st z( c,n (A) 8[(c C c RL(mC 4C 5in Z(mc,+n7CCl)) Vcc 2) R z 4 5L z (7nZ ct t C + a L(a ?( nc aci) i z x 5sizt rtCne) Z'4 C 4nc LnC j 2 {.j (c. C2(c3a4 4)2 mCe si 2Cn)3,) 4 L.rt C3R + R 3inzn4(+22 ?r C4 4 4 (C3+4) C4 Fi x L(IC3 ncC4) X sin ( C3n c4)( )+ I( C, in (,) in (,+s3 (nc (mCf c5n (mc3nc4) ( ) 2 (CA +.C4) C x S7csin (mc _) 3C4)( L(mc~ +nc4) 5 (n tn C4) L C 3 37 C4 L 4C ) f )l + bb (C3 ,C4)2 C L 5 mc(nc 3 c3nc4) () L(3m~c3?zc4) R (lnc) 4 C2 L(m c 3+n c4) 4) ] 4 C3 2 + (f3+ 4tC4) s2 2(Me+n ) R xn ce 2 (bZ RC 2L(Z n C+n e4) 3C4 R ) 2 44 Atl, C4) L( SiX 2(, C.3n'?C4F)( 4 2 V PR Si 471 C4 . ILC 4 R jI Ri R siz(m cnc ZL(mCz+tC) R +2L(meC C) 57 L z(t cf C ?) 2 t mcK z X in Zmc. \ZL77C, fni C3 m cDt *+2 RgL ,l 2?Lnc, sin 4?C g(C + SiYlZ ( eZn ( )4) 1 ?C2c:A $LL4?U 2 ^ 4 LflTC!L^ ^ x t Cze)(CZ*c1 bnc si Smc nc,) (L) (cz ,a ^ d L ) L(mc,C,) sin (m c1n c) a CaTU L (m x+n gy x5i(m +n c,) (C,  ae ( c nC C, 3 n Ce+7C) +2 C, CC [(C, c) (c2,C, (3m CL+4 f sin (i3" C ,+ X) + sxc) (iAc.(? 7t C2) () ( aCaC,)(c+ C)(m n ) x X5Cn (a3C1,?Z)( L( C14 C) ^ ~ d~/ i '^Vft^L c^C,) xs51n(fWlC4XN4 2 FC,/ R3 R O_~ R x IL C4 sin (m c,+ )(}J 2L(MC nY C) 4 ff ftf SR[ R sn z( I8 2L (nC 3 + C4) C itn C4 ) R 2L(R s in m3gnc n'C :t4 )) .2 L(m C37 Cf) R 4 2mC3 R t(7n1 C3) L ' 3 niC ZL JnC4 Sin. 4n C4 (i) R. + i4 Ms5 nC4( +C3z C4 mC C_3 Z~2a 3 c 3c4 tkbZ b3 C 3 C4 (C, + C4) (C4 C) ') L (mL zcn c4) R1S;?t (&L L ( C 3 7n C4) +(c, C) ( a c ) nic,+n4) ( 4 s3n (m 3n C )) ()Ji+ CCz b X L. (M C3 3n C4) 3 x c3 z a c4) (c4 C,) . (3S 7Z C + c + L (mccnc,) n ) L (n mc3 ?t c4) A~ c. C4) ( c4 + i c)) x cs c4)(c2,7 eC) x sin L(mc+Y'C,)(4) , stinZmc nc c( a^ a C flC)() SAl4 1C2 C 2 R 1 2 L(7~LnCj cZ S2 4,m e,() x sin (m C,  C 4) (4 R T5A;" (cMC nc4) (1) + x . L(3mc3,nc) L(,c~ nc4) S (c(]nc) ) ]b( I3 (, f c+ nc4.) 2L (m c, x.snz( Czn c4) (i) + SZL , y S 2,7 ) Cf A)C (7n cln cW) "K^S APPENDIX B APPENDIX B COMPUTER PROGRAM I STIFFENED CYLINDRICAL SHELLS UNDER AXIAL COMPRESSION 1 READ INPUT TAPE 5, 10,EEG,CDQRtHPIIIlDIFJIJDJ 1F,KI,KDOKF 10 FORMAT (8F5.3,914) DO 50 K=KIKFKD Z=.OI*FLOATF(K) DO 50 I=IIIF,ID X=.O1*FLGATF(I) DO 50 J=JIJFIJO Y=.O01*FLOATF(J) YS=Y**2 XS=X**2 XQ=XS**2 CS=C**2 CQ=CS**2 US=D**2 DQ=DS**2 QS=Q**2 QQ=QS**2 RS=R**2 RQ=RS**2 HM=1.H HP=1.+H HMS=1./HM**2 SI=E*(CC+QQ+XQ*(DQ+RQ)+6.*XS*(CS*OS+QS*RS)+G*I((C+X*D) 1*(DX*C))**2+((CX*D)*(D+X*C))**2+((Q+X*R)*(RX*Q))**2 2+((QX*R)*(R+X*Q))**2)/2.)/4. S2=E*XQ*(DQ0RQ+G*(CS*DS+QS*RS)) S3=E*(CC+QC+G*(CS*DS+QS*RS)) Gl=1./(l.+XS)**2 G2=1./(9.+XS)**2 G3=1./(1.+9.*XS)**2 Al=3.*(1.+XS)**2/32.+SI A3=(2.*(2.+H)*(I.+Z)*GLZ/2.)*XS A4=HP*4.*XQ*(G1*(1.+Z)**2+G2*Z**2+G3) A5=HP*(I.+XQ)/16. Bl=3.*XQ/8.+S2 84=HPe2.*XQ*GI*Z**2 85=HP*XQ*.5*(G3+(l.+Z)*Gl) 86=XS*.25*G1 C1=3./8.+S3 C4=HP*2.*XQ*GI C5=HP*XQ*.5*(G2+1t.+1./Z)*G1) C6=XS*(Gl+(1.+H)/8.)*.25/Z H1=C5+C6/YA5 H2=B5+B6/Y D2=AI*H2B1IA5 D3=IG1+A3*Y+A4*YS)*H2B4*A5*YS F2=AI*(C5+C6/Y)CI*A5 F3=(G1+A3*Y+A4*YS)*(C5+C6/Y)A5*(.25+C4*YS) Wl=(D2*D3*Hl*HI/F3**2+F2*H2*H2/F3H2*H*(F2*D3/F3+D2)/ 1F3)*HMS W2=HMS*3*XS*P*(D2*03*H1*(HI+A5)/F3*2+F2*H2*(H2A5)/F 13(2.H2*(HL+AA5)+A5*A5H2*A5(H1+A5)*A5)*.5*(F2*D3/F3+ 202)/F3) W3=HMS*2.25*XQ*P*P*(D2*D3*(H1+A5)**2/F3**2+F2*(H2A5)* 1*2/F3(HI+Ai)*(H2A5)*(F2*D3/F3+D3)/F3)+D2*D2+(D3*F2/F 23)**22.*D2*D03F2/F3 DISC=W2**24.*WI*W3 IF (DISC) 50, 60, 70 60 XIR=W2/(2.*WI) X2R=X1R GO TO 80 70 S=SQRTF(DISC) X1R=(W2+S)/(2.*W1I X2R=(W2S)/(2.*Wl) GO TO 80 80 WRITE OUTPUT TAPE 6, 20, ZtX,tYXIRX2RPtWIW2,W3,E 20 FORMAT (10F8.3) 50 CONTINUE GO TO I END 84 Symbols Used In Computer Program I E = 'ElJ G = K C = C1 D = C2 Q = C3 R = C4 H = P = p Z = X x ' Y = X2R = c COMPUTER PROGRAM II STIFFENED CYLINDRICAL SHELLS UNDER BENDING OR TRANSVERSE SHEAR 1 READ INPUT TAPE 5, 10,E,G,C,D,Q,R,HP,IIIDtIFJI,JD,J IF,KIKDKF 10 FORMAT (BF5.3,9I4) DO 50 K=KI,KFKD Z=.0OIFLOATF(K) DO 50 I=IIIFID X=.01OFLOATF(I) DO 50 J=JIJF,JD Y=.001FLOATF(J) ZS=L**2 YS=Y**2 XS=X**2 XQ=XS**2 CS=C**2 CQ=CS**2 DS=D**2 DQ=DS**2 QS=Q**2 &Q=US**2 RS=k**2 RQ=RS**2 HM=1.H HP=l.+H HMS=1./HM**2 S1=3.*E*(CQ+QQ+XQ*(DQ+RQ)+6.*XS*(CS*DS+QS*RS)+G*((IC+X I*D)*(DX*C))**2+((CX*D)*(D+X*C))**2+((Q+X*R)*(RX*Q)) **2+((QX*R)*(R+X*Q))**2)/2.)/8. S2=1.5*E*XQ*(DQ+RQ+G*(CS*DS+QS*RS)) S3=1.5*E*(CQ+QQ+G*(CS*DS+QS*RS)) Gl=l./tl.+XS)**2 G2=1./(9.+XS)**2 G3=i./(l.+9.*XS)**2 A1=9.*(I.+XS)**2/64.+S1 A3=(1.5*(2.+H)*(1.+Z)*Gl+3.*Z/8.)*XS A4=HP*9.*XQ*(Gl*(I.+Z)**2+G2*ZS+G3)/4. A5=HP*9.*(l.+XQ)/256. 81=9.*XQ/16.+S2 b4=HP*9.*XQ*G1LZS/8. 65=HP,9.*XQ*((l.+Z)*Gl+G3)/32. L6=(3.*XS*GL)/16. CI=9./16.+S3 C4=HP*9.*XQ*G1/8. L5=HP*9.*XU*((1.+1./Z)*Gl+G2)/32. C6=3.*XS*(HP/8.+Gl)/(16.*Z) HI=C5+C6/YA5 H2=B5+B6/Y 02=A1*H2B1*A5 03=(G1+A3*Y+A4*YS)*H2B4*A5*YS F2=AI*(C5+C6/Y)CI*A5 F3=(G1+A3*Y+A4*YS)*(C5+C6/YlA5*(.25+C4*YS) W1=HMS*.25*(D2*D3*HI*Hl/F3**2+F2*H2*H2/F3HIlH2*(F2*D3 I/F3+021/F3) W2=HMS*1.5*XS*P*(D2*D3*Hl*(HL+A5)/F3**2+F2*H2*(H2A5) I/F3(2.*H2*(HI+A5)+A5*A5H2*A5(HI+A5)*A5)*.5*(F2*D3/F 23+D2)/F3) W3=HMS*2.25*XQ*P*P*(D2*D3*(Hl+A5)**2/F3**2+F2*(H2A5)* 1*2/F3(Hl+A5)*(H2A5)*(F2*D3/F3+D3)/F3)+D2*D2+(D3*F2/F 23)**22.*D02D3*F2/F3 DISC=W2**24.*W1*W3 IF (DISC) 50, 60, 70 60 XIR=W2/(2.*WI) X2R=X1R GO TO 80 70 S=SQRTF(DISC) XIR=(W2+S)/(2.*W1) X2R=IW2S)/(2.*Wl) GO TO 80 60 WRITE OUTPUT TAPE 6, 20, Z,XYXIR,X2RPtW,W2,W3,E 20 FORMAT (10F8.3) 50 CONTINUE GO TO 1 END Symbols Used In Computer Program II E = El 7 C = C1 Q = C3 H = F Y = f G = K D = C2 R = C4 P = p x = X2R = 2 2 = 2(b + 3/2') or (X2R = 2 3 = for transverse load) BIBLIOGRAPHY 1. W. Fairbairn, "On the Resistance of Tubes to Collapse," Phil. Trans. Roy. Soc., Vol. 148, pp. 389413, 1859. 2. G. H. Bryan, "On the Stability of Elastic Systems," Proceedings of the Cambridge Philosophical Society, Vol. 6, Part 4, pp. 199211, 1888. 3. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, fourth Edition. Reprinted in the U.S.A., by Dover Publication, New York, N. Y., 1944. 4. S. Timoshenko, and J. Gore, Theory of Elastic Stability, McGrawHill Book Company, New York, N. Y., 1961. 5. R. V. Southwell, "On the General Theory of Elastic Stability," Phil. Trans. Roy. Soc., Series A, pp. 187213, 1914. 6. L. H. Donnell, "Stability of ThinWalled Tubes under Torsion," NACA Report 479, 1933. 7. S. B. Batdorf, "A Simplified Method of Elastic Stability Analysis for Thin Cylindrical Shells," NACA Report 874, 1947. 8. H. S. Suer, and L. A. Harris, "The Stability of Thin Walled Cylinders under Combined Torsion and External Lateral or Hydrostatic Pressure," Journal of Applied Mechanics, Vol. 81, Series E., No. 1, pp. 138139, 1959. 9. P. Side, and V. I. Weingartern, "On the Buckling of Circular Cylindrical Shells Under Pure Bending," Journal of Applied Mechanics, Vol. 28, No. 1, pp. 112116, March, 1961. 