EP Iabagea' a (Es — Yeo Lt ( R LL? Ly Ay

VIBRATION OF STRESSED SHELLS OF DOUBLE CURVATURE

by -

Paul Aw Eboper

Dissertation submitted to the Graduate Faculty of the

Virginia Polytechnic Institute

in candidacy for the degree of

DOCTOR OF PHILOSOPHY

ENGINEERING MECHANICS CN = | APPROVED: L/ en’ F v7 dork Chairman, Prof. Daniel Frederick

1¢ imistion tang dee \hoe aw

Prof. R. L. wate Prof. H. 1, Johhéon EP iabagea' a (es —_ Yeo lt ( r LL? Ly Ay

Prof. V. Maderspach Prof. R. McNitt

Blacksburg, Virginia

June 1968

4 LD 5655 Y856 19628 Coz

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eels eels re re II. TABLE OF CONTENTS

CHAPTER PAGE

IT. TITLE . 1. 1 1 ww ww ww we we ew we ee ew ee ee

II. TABLE OF CONTENTS . 2. 6 2 © 2 © ee 8 eo we ee il

III. ACKNOWLEDGMENTS . 2. 2. 6 6 2 © © ee ew ew ew ew ws iv

IV. LIST OF FIGURES AND TABLES . 2. «© © «© © © © we

V. INTRODUCTION . . 2. 2 © © © © © © © ew ww

VI. SYMBOLS . 2. 2 2 2 6 0 ew ew ew we ew ew ew we

VII. NUMERICAL ANALYSIS OF VIBRATIONS OF GENERAL PRESTRESSED SHELLS OF REVOLUTION . . 6 2 6 © © © © «© © we ww

A. Development of General Equations of Motion

1. Governing nonlinear equations ...... 11

2. Vibration equations . . 2. « « « «© «© « © «© « e 16

4%. Reduction to ordinary differential equations. 2d

B. Closed Form Solution for Cylinder Vibrations 26 on C. Development of Numerical Solution ...... e.

1. Development of difference equations .... et

2. Numerical solution ...... -. «© «© «© » 2 eo 32

VIII. APPROXIMATE SHELI EQUATIONS . 2. 2. 2 6 2 © © © we we we wo a9

A. Derivation of Approximate Shell Equations ... D9

B. Solution of Approximate Vibration Equations . OL

C. Solution of Approximate Vibration Equations for a Freely Supported Shell .....e 2. «es 60

D. Membrane Solution of Approximate Vibration

Equations e e e e Cd o e Sd e * . e e e e ° e e ° 61

ii iii

CHAPTER PAGE

1. Solution of membrane vibration equations for a freely supported shell ...... 63

2, Characteristic roots from membrane theory .. 65

TX. RESULTS AND DISCUSSION... . 2 ee ee wee ee eee 69

A. Shell Configurations Investigated ...... +.- 69

B. Accuracy of Solutions « « «6 2. +e © © © ee we ew © [2

C. Effects of Meridional Curvature ...... 80

D. Membrane and Pure Bending Analysis ...... 85

E. Prestressing Effects of Lateral Pressure ..... 90

F. Effects of Edge Restraint ...... -e ee eee

X. CONCLUDING REMARKS . 2. 2 2 6 © © © © @ © © we th ew ew ew ew et OD

XI. REFERENCES . 2. 2. 2 2 0 © © © © © © © we ow we ew ew we hw wl el eC LOL

XII. VITA. www we ee we ee ww we tw ww we we tw ww we LOK

XTII. APPENDIX A. 2. 2. 2 2 0 6 © © oe ew ew ew ow tw we ww ww ww ew LOD

XTV. APPENDIX B... 2. 3 2 1 2 0 © ee ew ew ew ew ew ew ew ew ew wl ew LID

XV. APPENDIX C . 2. 2 6 2 ew ee we ew ew eo we we ew ew we ew LY

XVI. APPENDIX D. 2. . 2 0 ew ww ww ew we we et wt eh wh eh whe LLL

XVIL APPENDIX E ° » ° e e e e s e e J ¢ e e . e cf se e e 6 e ° 124 IIT. ACKNOWLEDGMENTS

The author wishes to express his thanks to the National

Aeronautics and Space Administration for permitting him to perform this research as part of his work assignment at Langley Research

Center. He also wishes to thank Dr. Daniel Frederick, Professor of

Engineering Mechanics, Virginia Polytechnic Institute, and Dr.

Manuel Stein of NASA, for their council and guidance, and Mrs.

Nancy Sykes, Mrs. Martha Robinson, and Mrs. Cornelia Dexter for developing the computer programs and finally, his wife Marilyn for her patience and consideration throughout the course of the research.

iv IV. LIST OF FIGURES AND TABLES

FIGURE PAGE

1. Shell middle-surface geometry . .. 6. 6 «© «© © © © © «© «© © IO

2. Middle surface quantities . .. 1. 1. 6 26 «© 6 «© © © © © ew © LD

(a) Stress resultants and displacements ...... - 15

(b) Moment resultants and rotations ...... - 15

3. Geometry of class of shelis investigated ...... +8 YO

4, Comparison of minimum frequencies from approximate analysis with those from,numerical analysis for freely supported shells (5, = 40.053 5S = 3;

a= =_ 10001000) ° ° e ° ° o ° ° e e e e ° e . . e e e ° . e e {34

5. Comparison of minimum frequencies from approximate analysis with those from numerical analysis for freely supported shells (ky, = 20.1; S = 3; R 4 = 1000 } e e ° ° ° . e ° ° e . ° ° e ° e . . . e e . «6 Th

6. Comparison of minimum frequencies from approximate analysis with those from numerical analysis for freely supported shells (,, = 40.15; S = 3; R 4 = 1000 ) ° ° . . « ° . e@ ° ee 08 « e © @ » o e . ° e e «6 15

{. Comparison of w meridional modes for several n for successive degrees of curvature, e = -0.05, -O.1, -0.153 3=3; F=1000).. 6.00.0... .0006. 77

8. Comparison of minimum frequencies from approximate analysis with those from numerical analysis for freely supported shells (x, = 10.05; S = 10; 5 = 1000). 78

9. Meridional modes of w for several successive n (k, = -0.05; S= 10; R =1000) ...... 79

iO. Effect of meridional curvature on the fundamental frequencies of freely supported unstressed shells (+ = -) percent to +5 percent; S = 3; - = 1000] oe « « 82 vi

FIGURE PAGE

ll. Effect of meridional curvature on the fundamental frequencies of freely supported stressed shells for various lengths and thicknesses (1 = -5 percent to +5 percent; S=1, 5, 10; : = 100, 200, 1000 ) oe 84

1e. Effect of meridional curvature on the natural frequencies of freely supported unstressed membrane

shells (sy = ~0.5 to +0.5). ° « . e e ° ° e ° e ° e ° 86

13. Effect of meridional curvature on the fundamental frequencies of freely supported shells subjected to a constant circumferential tensile and compressive prestress (+ = -5 percent to +5 percent; S = 3;

h =— 200)200 e e e e e ° e ® e e e * ° * e es s e e e e e . 91 14, Pressure stabilization of freely supported shells (+ = -5 percent to +5 percent; S = 3; - = 1000 } o 8 99

15. Effect of constant circumferential prestress on the fundamental frequencies of freely supported shells T = -3 percent to +3 percent; S = 3; A = 200; Ne -4 ~4 yp =72 x10 to +2x 10 ceeeh ee 99

16. Effect of edge restraint on the natural vibration of unstressed shells (x, =+0.1; S = 3; : = 1000 } 2 8 96

TABLE

Comparison of characteristic roots obtained from membrane and bending theories (n=4)...... 89

it. Comparison of the effect of edge restraint on the natural frequencies of a negative curvature shell R (sy = -0.05; 5 = 45 a => 1000 | o «© © e «© © e© # «@ 97 V. INTRODUCTION

In the design of shell structures for launch vehicles, planetary atmospheric entry probes, or similar structures, knowledge of the natural frequencies and mode shapes of the systems is of fundamental importance in determining their dynamic behavior. Shells of double curvature are common structural elements in aerospace and related industries, but due to the complexity of their configurations and governing equations, little has been done to classify their general dynamic behavior. Reference 1 gives a survey on the present state of the art in the area of analysis of free vibrations of shell structures. The subject of this dissertation is the determination of the effect of the meridional curvature on the natural vibrations of a class of axisymmetrically prestressed doubly curved shells of revolution. Two methods of approach are used in this analysis. [In

Chapter VII the exact equations of motion are solved using approximate techniques of solution, while in Chapter VIII approximations are made to the exact equations so that closed form techniques of solution are possible. Both of the above methods are used in this investigation to complement one another.

Since the shells are too complex to allow closed form solutions of the exact equations, numerical techniques are used in Chapter VII to analyze the dynamic behavior. Several numerical methods have recently been developed for static stress analysis and unstressed free vibration analysis of general shells of revolution. Static stress analysis is considered in references 2 to 5 and unstressed vibrations are considered in references 6 and 7. Membrane and flexural vibrations of toroidal shells are treated in references 8 and 9 on the basis of the numerical approach of reference 2. In the present investigation, differential and difference equations that govern the asymmetric linear vibration of a general class of prestressed shells of revolution are derived using the nonlinear shell theory of reference 10. A finite difference procedure similar to that of reference 2 (utilizing and extending the ideas of references 8 and

9) is then formulated to obtain solutions of the equations.

Numerical methods yield accurate results; however, it becomes impractical to attempt an extensive parametric study of the behavior of shell systems even with high speed computers. Rapid closed form techniques of solution are possible when approximations are made to the equations of motion; however, the range of application is restricted. These techniques are used in Chapter VIII. The solutions are limited to shells of revolution with shallow constant meridional curvature with no shallowness limitation in the circumferential direction.

By perturbing the geometry of the cylinder and referencing the zero strain state at this new configuration, a set of nonlinear homogeneous field equations governing the dynamic behavior of doubly curved shells with positive or negative constant shallow meridional curvature is developed. The procedure used in developing these equations is similar to that used in references 11 and le where

Donnell-type equations are developed from flat plate equations. The nonlinear cylinder strain-displacement relations from reference 10 are modified slightly in the twisting curvature relation and small

nonlinear terms are deleted from the middle surface strain relations.

The linearization and geometric perturbation of these equations

coupled with the assumption of constant prestress rotations result in a set of strain-displacement relations which have corresponding equilibrium equations with constant coefficients. These equations govern the natural vibrations of prestressed nearly cylindrical shells of revolution and are solved here for the natural vibrations of shells with freely supported edges similar to that which was done previously for cylindrical shells, see for example references 13, 14, and 15.

Results from the shallow shell analysis of Chapter VIII are

compared with those obtained by the use of techniques and equations

of Chapter VII to indicate the range of application of the present analysis. This simplified analysis is then used to investigate the effects of meridional curvature on the fundamental vibrations of freely supported shells as the length and thickness parameters are varied.

The equations described above are also solved for the natural vibration of membrane shells and inextensional shells with freely

supported edges. The characteristic roots of the membrane field equations are also investigated and compared with the corresponding characteristic roots of the bending field equations to show the dependence of frequencies on shell membrane resistance. The solutions to the membrane theory afford a simple method of determining shell configurations of negative curvature for which bending action pre- dominates during vibratory motion in certain modes.

The approximate analysis is then used to investigate the effects of meridional curvature on the fundamental vibrations of shells as the length and thickness parameters are varied, and to determine the degree of stabilization afforded by the application of constant directional lateral pressures to the shell systems. Finally, the approximate solution is used to determine the unstressed fundamental frequencies of clamped shells and to determine the stiffening action due to the additional edge restraint. Me Ne N n Z| oO QO g7 Ng sMg

Al 8 My Ne gsNggrQp2Qy goMg, Young's bending moment modified modified modified nondimensional reference functions extensional displacement number nondimensional in-plane central number total shell station eqs. perturbed perturbed (eq. number thickness VI. variable of of (19)) (75)) modulus rise number, stiffness moment displacement prestress stress length defined axial circumferential SYMBOLS stiffness state state of amplitude of prestress curvatures half-waves of the resultants i intervals resultants (see by = (eq. elasticity stress 0, shell equations eqs. amplitude (eq. 1, coefficients (16)) parameters meridian waves 2, along resultants (20)) (eqs. associated associated (16)) «++, (77) the coefficients (12)) N meridian (see with with surface loading

nondimensional radius of cross section, p/a

radius of cylinder

principal radii of curvature (see fig. 1)

total meridional arc length

total nondimensional meridional arc length,

s/a; s/R

time

displacement variables in meridional (&),

circumferential (8), and normal directions,

respectively, of undeformed middle surface

defining perturbed state

nondimensional meridional coordinate

ratio of the circumferential to axial wavelength

length of interval between stations, S/N

modal normalizing factor

cofacters of 4 x 4 matrix given in eq. (51)

middle-surface strains associated with the

perturbed state

MD MD circumferential coordinate in undeformed shell

KerKo keg middle-surface bending strains associated with

the perturbed state

characteristic roots

thickness parameter, a ; rol rol

Poisson's ratio

mass density

meridional coordinate in undeformed shell

cross-sectional radius (fig. 1)

percent ratio of meridional rise to length

middle-surface rotations associated with

perturbed state

prestress meridional rotation

natural frequency 2 am ay (1 - *)

Q7 = z frequency parameter

Notations Used to Identify Load and Deformation Variables:

Unmarked variables indicate variables associated

with perturbed state only

indicates modified variables associated with

the total deformation measured with respect

to an unstressed shell

“N indicates variables associated with total

deformation measured with respect to an

unstressed undeformed cylinder

indicates modified variables associated with the

prestress state only

indicates physical stress resultant quantities The comma before a subscript denotes differen-

tiation with respect to the following

subscripted variable. A dot over a symbol

indicates differentiation of the quantities

with respect to time.

1x 4 column matrices:

Z. dependent variable i 1 x 8 column matrices:

A. amplitude coefficients J 4 x 4 matrices:

A, 2B, sC, sDysDyrEy5B N difference-equation coefficients Cs ts, boundary-equation coefficients

equilibrium-equation coefficients Ba Oj? Ay

recursion coefficients a,8 boundary-condition-selection matrices

8 x 8 matrices:

Noi boundary stress coefficients

Yay boundary displacement coefficients

2, boundary-condition-selection matrices VII. NUMERICAL ANALYSIS OF VIBRATIONS OF GENERAL PRESTRESSED * SHELLS OF REVOLUTION

A. Development of General Equations of Motion

In this section, the linear equations of motion are derived for a symmetrically prestressed shell of revolution with an arbitrary meridional configuration. The nonlinear shell theory given in reference 10 forms the basis for all analytical work done in this and subsequent sections.

The shell geometry is illustrated in figure 1. The location of points on the middle surface of the shell is described by the principal coordinates (&,9), where & is the meridional distance measured on the middle surface from one boundary, and 98 is the circumferential angle. Since the shell is axisymmetric, it is completely described by the meridional shape parameter o(&) which is the radial distance from the axis of revolution to the middle surface of the shell.

The principal radii of curvature of the middle surface, R,(§) and Ro(), are given by: 1 -(@e) ap ae=

¥ Most of the material covered in Chapter VII has been pre- published in the report "Vibration and Buckling of Prestressed Shells of Revolution" NASA TN D-3831, March 1967. 10

Normal

“2 Middle surface

Figure 1.- Shell middle-surface geometry.

11 | Jo)

The shell is assumed to have a constant thickness h measured along the normal to the middle surface and boundaries at §&€ =O and

€ = s, where s is the total meridional arc length. The material is assumed homogeneous and isotropic with mass density v, Young's. modulus of elasticity E, and Poisson's ratio u.

1. Governing nonlinear equations.- General nonlinear shell

equations in which strains are assumed to be small and rotations, moderately small, are given in reference 10. For a shell of revolution, these equations become, when inertia terms are added,

~ ~ do ~ Oo 1/1 1 \~ (of, | é + Neove 7 ae Ny + Re Qe + 3 ( - = Mea .e

? 7

- Gar . ( Pat¥o , + pP = phvW (2c)

? 3

(oii, + M _ © H - of, =0 (2a) E £ 60,9 d& 8 E 3

oM &0 +M 0,0 °+ 2Rgd—~€9 eQ, 8 = 0 (2e 26 ,

where the comma before a subscript denotes partial differentiation with respect to the succeeding subscripted independent variables

(€ or 98) and dots over a quantity denote differentiation with respect to time. The equilibrium equations (2a), (2b), and (2c) represent the sum of forces along coordinates of the undeformed surface, and the perturbed displacements of the middle surface JU,

V, and W are measured in the direction of the coordinates of the undeformed surface with W measured positive along the outward normal.

The boundary conditions considered on the edges € = 0 and

€ =s may be chosen from any combination of the following four pairs of quantities in which either quantity (but not both) of each pair is prescribed: a)

Py or Me ;

The equations have been derived by use of the Kirchhoff-Love assumptions, that is, normals to the undeformed middle surface remain normal to the deformed middle surface, normal strain is zero, and the normal stress is negligible. Rotary inertia terms have been neglected in the moment equations. Modified stress and moment resultants have been used in the development of equations (2) and (43) and are defined as follows: \ Oo Ne = Np - R,

Oo M N. = N -— 8 9 Ry

Oo ~ _y Oo] Moe

g6 ~ “£8 Ro

Oo N =N Oo] Meg 6& “GE R E

~ Oo M, =M E E 14

| 0 | U = mae | | ~ Ne L oO Meg == Mog = = 3 (Meg + Moe } | i (4)

The modified transverse shear stress resultants Q, and Qq may be found by applying the definitions in equations (4) to the moment equilibrium equations (eqs. (2d) and (2e)). O The quantities Neos Neos N Nog» Qe and Qa represent go ? the total middle-surface stress resultants (fig. 2(a)), and the quantities Mes Ma Megs and Moe represent the total middle- surface moment resultants (fig. (2(b)). No attempt is made to relate these stress and moment resultants to the distribution of stress through the thickness of the shell. The equations of reference 10 have been derived without dependence on such a relation- ship; thus, any formulation consistent with thin-shell theory is acceptable. According to reference 10, the addition of terms iike

M°/R to the N° quantities in the stress-strain relations does not introduce errors any greater than those already introduced by neglecting transverse shear flexibility in the Kirchhoff-Love hypothesis. Consequently, the N quantities may be treated as stress resultants without introducing an inconsistency in the thin-shell analysis.

The sum of the moments about the normal direction is M 0 0

Neg o_ 88Ry Ly GE Oo 4 R$9 _ 15

Normal

(a) Stress resultants and displacements. cb.Normal

3 hm 9 ee E (Du 9 9 g

(b) Moment resultants and rotations.

Figure 2.- Middle surface quantities.

Hence, from the definition of the modified stress resultants sr sr

Noe = Neg mee em em

Therefore, the sixth equilibrium equation, that of equilibrium of moments about the normal, is identically satisfied by symmetric modified stress resultants.

2. Vibration equations.- In the derivation of the vibration

equations, the total rotations and total stress and moment resultants are separated into the parts associated with the initial axisymmetric prestress and the parts associated with the infinitesimal perturbed time dependent displacements about the prestressed state. Symbols with bars represent those quantities associated with the prestress conditions and symbols without bars represent those associated with the perturbed state. Thus, the total stress state may be completely described by these quantities from the relations:

N, = Ne(€) + N,(8,9,¢) .

Ng = No(£) + No(£,9,t)

Neg = Neg (£59,t)

Q, = O,(€) + Q,(8,9,¢) \ Qy = %(E,9,t) 17

rae rae = Ti, (£) + Mp(£,,¢) i i

ae ae My(€) + Mg(&,9,%)

rae £9 = Mp g(£»9,+) | (6)

The total rotations are

= ] ,(€) + ,(6,9,%) 3 une

= ! Pg \ €,9,t) (7) Be

@ = o(£,9,t) and the total surface loading is described by

H re B,(t) (8)

tl De P(E)

since no additional surface loading is assumed to be associated with the vibrating state. Substitution of equations (6), (7), and (8) in equation (2a) yields

oeae Fve Lp kayade ~ ae *o* +£R, Q Oe -27 Ry 56,5MeN +P P, +1 SP Ne ++ ONoN

dp p l/l 1 p *Ngo,o de No * Re * a(R -iy ) "42,0 “oR, Pee 18

op — Pga Peg - 5kh c (Ne + *) \ + ‘ a 7) gE 9 . > = ~ =\l_ vs { my ON, - 5% 4 (Ne + nh = pvhU (9)

where Q-derivatives of barred quantities vanish as a result of

axisymmetry of the prestressed state. The prestressed shell is in

equilibrium, thus, the sum of the terms enclosed by the first set of braces in equation (9) vanishes identically. Furthermore, the perturbation of the shell away from the prestressed configuration is

governed by linear theory. Therefore, the nonlinear terms enclosed by

the second pair of braces are neglected.

The term in equation (9) enclosed by the third pair of braces

(i.e., the interaction between the prestress deformation and the perturbation stress resultants) is usually neglected in the procedure followed in the classical linearization process for a cylinder. If

this term is neglected, the general assumption is made that the prestress rotation is uniformly zero throughout the shell. The error

introduced is usually negligible, but for certain boundary conditions

or for sharply varying surface loads, this term may be significant and is consequently retained in this analysis. If oe and Pee are neglected, the problem may be reinterpreted as that of a pre-

stressed but undeformed shell of revolution perturbed about the

undeformed state. With this term retained, the equilibrium equation

in the meridional direction (eq. (9)) reduces to 19

do p ( Ong b)| + Nig£0,0 9 - aE d& NG8 t Re 003)1f/_ (F -LR Meg 8a, Ne: Q

-51/5. (Ne + +5Ng) Pg _28 ON.ON, == phvwphvu (10a)10

By the same procedure, the remaining equilibrium equations

(eqs. (2b) to (2e)) are linearized, and reduce to:

- Re NgPq +5 (% + >| - Re 8,Neg = ohvV (10b)

Pe) + Q g-P f e, + =|8\ - (Pee)(= - Nog= 9 - (PM) =

- OeNegg = phvw (10¢)

(me) + Meg 9 - se M, - pa, = 0 (10d)

Meo) +My9 + SE Meg - P% = 0 (10e)

Solving equations (104) and (10e) for Qe and Qo and substituting for them into equations (10a), (10b), and (10c), eliminates these quantities from the system. The parameters defining the geometry of 20

the middle surface can be nondimensionalized by using a reference

length a, as follows:

x= a (12)

r= a and nondimensional curvatures can be defined as

a kK =-=— x R Me oe (12) 8 Ry

Upon completion of these manipulations, the following equilibrium

equations result:

dr dr ar dr a(S Ny + TNs x + Neg,e - ax Ne) + kL ax Me + THM, x - kK ax My

+i 3k -k,\ M -a|rk NN. +i(N, +5 D 2 x 6} ~£0,8 ere Ove 6} *,8

+ rk_O,N, | = ra hv (13a) xe &

NMI el

2 d dr d -ar Gar + KN +=, +2 ax M + rM - a € Ex * E Xx

OU ax 1

+ M g0,0 * Meo xo + > Me a0 * (- KEN RIS

Bi E — dr — 8 ~ F—~ Oe - Noo 9 ~ Gx Pee 7 R

— Om - FNgo9 ) > ra hyW (13c)

Similarly, the boundary conditions (eqs. (3)) are given as

Ne O or Us=

1 lL f= = Neg + 3e (3g - i.) Meg + 5 (Ny + No) ? or

1 1 dr £6,080; — — oO a a * 5 a (Me - Mo) * |. NePg ~ Pee II or

i: or

The modified stress-resultant—strain relationships, if physical linearity is assumed, are Ny = B(e + <9) Ng = B(eq+ Hey) Neg = BIL - u) gg Ny = D(my + tg) 22

2 Mo = D(kg + Hig) = WM, + D{1 - wu } Kp

Meg = D(1 - pu) Keg (15) where

Eh B= l- um (16) D= —_ 3 12(2 - i )

The linearized strain-displacement relationships, from reference 10, reduce to

‘\ cpg 1 (Ux tM + _TAU - _OM)

V 1jidrwv 8 =siu{joe_. —2— £9 a & r + r + ay

1 U 9 V x drv Q E Q E =| uw22 2 OO —. - — “e¢ a (2. +3 dx Br + B Kev er We)

1 ds. me = a Ux te U- V xx (17)

_ oi fay, Sey | 00 ar x 6” 2\a&r r ,0° #O "dr. a r 22 BK - a) _ CX -*) L& = 3k) ar |

_ W 1x0 , dr W *,8

r + ax 2 r ~

The middle-surface rotations are given in terms of displacements as

5 1 (x0 - Wx)

W 9

® = = oy -— (18)

U _1l dr V 39 o-b(v +E. 2]

The prestress terms are given nondimensionally as

Ne = Be, (x) Ng = Be, (x) (19)

4. Reduction to ordinary differential equations.- With equations

(13) and (15) to (19), the equilibrium equations can be reduced to three partial differential equations with the displacements as the unknown dependent variables where the highest order derivative in x is a fourth-order derivative. However, since the solution, in general, can only be achieved by numerical techniques, the procedure of reference 2 is followed, where dependence on 9 is removed by assuming a solution of the separable type and introducing M, as an additional unknown. This procedure yields a set of four second- order ordinary differential equations with variable coefficients.

The fourth equation is simply the equation for M, E in terms of the displacements. This reduction in order is essential for the numerical treatment that follows. ey

A solution is assumed of the form

~ U = u(x) (cos né) ei

V = v(x) (sin n@) ett (20) W = w(x) (cos n@) elt

M,(72= =~Eh? m,(x)m,(x) ( (cos n@) 8) e Lot |

Defining the perturbation displacements in this manner assures compatibility in the 9-coordinate.

Performing the operations indicated and utilizing the following geometric relationships

ae g * 51 aedr (& > *s)

(21) 2

which are the Codazzi and Gauss equations, respectively, yields the governing equations, as follows:

Fie tf + G,4u t + Hu + G3 oY t + HL ov + BW '1 + Gj3W { + HW

+ Gym, | + Hy My = 0 (22a) 2

Gou + HU + Fyo¥ tt + Gov $ + HooV + Poa 14 + GozW too HW

+ HM, = 0 (22b)

Fay¥ ft + Gz4¥ to HeyY + oY tty Grav Soo4. Ha pV + F 33% a a 33 w'

+ Hy sw + Faye tf + Gym, 5 + Hj My = 0 (22c)2

Gul + Hu + Hyov + Fy aw" + Gy gw + Hy 3W + Hyym, = QO (22d)

The same procedure yields the boundary conditions, as follows:

! q 2 iW iW

e,,ul + fjjut fv + e,3¥ + fw i O or u=QO (23a)

oO oO

i i

ut ut

< < 4 4 f.u+e_¥wyv'21 22 +f4v22 +e 23 w' + fiw23 © 0 (23b)

Cn! + fu + CzoV" + fo + en5W + 330 + &5)My + Foye = 0

or w= 0 (23c)

fy jut ey 3W" =O or m, =0 (23a) E

Primes denote total differentiation with respect to the nondimensional variable x, and the coefficients are subscripted for convenience in subsequent matrix manipulations. The coefficients F G H jk? “jk? “Gk are given in appendix A in terms of the parameters 7Y and A, where 26

(24)

the frequency parameter of occurs in Hy Hoos and Teas where

22 2 2. Swit 4!) (25)

B. Closed-Form Solution for Cylinder Vibrations

The vibratory characteristics of a "freely supported" cylinder

(simply supported but unrestrained in the axial direction, i.e.

Ne =Ve=We M, = 0) with prestress deformations neglected are well

known. (See refs. 16, 17, and 18.) Thus, these known results can be used as a check of the validity of the governing equations and

of the accuracy of the numerical techniques to be suggested

subsequently.

When prestress deformations are neglected and the in-plane

stresses are constant, the equations (22) reduce to ordinary differ-

ential equations with constant coefficients which, for freely

supported boundary conditions, have a solution of the form

mmx u(x) = A, 08 =

v(x) = B, sin = (26) wlx)(x) = C, sin mie

m,(s) =D, sin >max m= 1, 2, ««. ey

where S is the length-radius ratio of the cylinder (a = cylinder radius). The classical procedure of neglecting prestress deformations to ensure constant coefficients in the field equations implies that the cylinder is initially prestressed as a shell with free edges and then subsequently supported for vibration.

Equations (26) are substituted into equations (22) to yield a

set of linear homogeneous algebraic equations. For a nontrivial

solution to exist, the determinant of the coefficient matrix of the resultant set of equations must vanish. This procedure leads to the

characteristic equation

,(a®) + a,(a°) + A, (9°) + Ay = 0 (27) where the coefficients A, are given in appendix A.

C. Development of Numerical Solution

i. Development of difference equations.- A numerical procedure

is needed for those shells of revolution and loading conditions which do not admit a solution in closed form. The meridian of the shell is divided into increments and a three-point central difference method

is used to reduce the differential equations to algebraic form. The

distance measured along the meridian between adjacent stations is

constant and is represented nondimensionally by A where

(28) 28

and where the subscript i on symbols and matrices indicates the

evaluation of the subscripted variable or matrix at the ith station,

i =0, 1, 2, ***, N and where

S total meridional arc length of the nondimensional shell, s/a

N total number of intervals

The three-point difference formulas, when applied at the ith station for some function 2z(x), are

(29)

Reference 43 indicates that this simple approximation leads to suffi-

ciently accurate results.

The governing equations (22) may be written in matrix form at

station i as

F,Z,'' + G,Z,' + HZ, = 0 (30) where

[" Fy O Fis Oo |

| 0 Foo Fos O

Bo = r (31a) 31 P30 F353 *5 | o O Fis O | eg

[ Gy Gio G3 Gay,

Gay Goo Coz 0 G, = (31b) Gy Cx Cas Cr, [Sia ° Cys os,

at ho 5 Hy

Hoy Hoo Hos Hoy H, = (31c) ty Tso Has Foy Ai Typ His | = a, and y Ze = (414) W m B) 4

Similarly, all the boundary equations (243) may be written ir

matrix form at the boundaries, 1 =0 and i=N, as

Lye_%g | + (cnt, + Bo) Zo i oO oO (32) Oye yay’ + (Agfy + By) Zz = © where 30

OQ 22 (33a) “oN = al 32 “sh

0 0 L Jon

TON ~ (33b)

and where 6 | 1% O 0 “o,N = 0 “33 0 0 “yl O,N re 5 7 (i - a4) 5 - 0 (1- O55) 0 O,N 0 O O (2 = 4s) J0,N |

take on the value 1 or O depending on the The elements a Jd prescribed conditions.

The a~ and f-matrices (eqs. (34)) are used to select the prescribed boundary conditions. If, for example, u = 0 is prescribed

at i-=0O then (11), =O andif u tis not prescribed at i=N

(i.e., if the u displacement is unrestrained in the meridional

direction of the undeformed shell), then (“11) =1. If desired, the present theory can be extended to allow for elastic and directional

supports in the boundary conditions by appropriate redefinition of

the a- and Bp-matrices.

When equations (29) are applied to equations (40) and (32), the

governing equations become

Wy i eF, Fy G, q2 bs

(i= 0, 1, 2, ---, N-1, N)

and the boundary equations become

%Kko Xoo - “BR 21 + (aipfy + By )Zq + aR =O (36)

BR Nn * (ayy + By | Zy + i Zany =

Equations (36) can be solved for Z and Zy,, ond the results -1 can be substituted into equation (35) to yield, at i = 0, e. of /F Fo Go % \rt OF 0 XH 3(2-2| fi, = 2) #2) #8 Ao

ae F G -1 F G +sO-O (330 - 2) 0 Beg} 0 0 Z, = 0 (37) 32

-1 | e FE G. or | N; N N N 1B Se By Bo) AN} PP

yey | f Fy Sy - By Sy

A A where

1 0 O 0

0 1 OQ O T= 0 0 1 0

Lo 0 0 1

The difference equations (35), (37), and (38) constitute a complete

set of field equations governing the behavior of the perturbed state.

2. Numerical solution.- The problem is now one of solving a

set of homogeneous equations (eqs. (35), (37), and (38)). This set

constitutes an eigenvalue problem such that the mode shape Zs is the eigenvector and the frequency parameter oF is the corresponding eigenvalue for the vibration problem.

The fourth equation, being simply the definition of ma in terms of displacements, will not contain an eigenvalue. For a nontrivial

solution to exist, the determinant of the coefficient matrix must vanish.

The coefficient matrix will be a 12-element-wide band matrix. A

convenient technique for solution of this problem can be formulated D3

by modifying the method of references 8 and 9. Such a modification

is presented herein to handle the free vibrations of a prestressed

shell governed by four second-order difference equations with two-

point boundary conditions.

Define the following (4 x 4) matrices:

F, G A «2+. i Ac oA

eF, B. = - — i i Ae

F, G C,=—=+—i 1 i A cA

e roa. oF D, =a, |arx|0 0s>-a-] 0 [H-—)+2,]/+8 0 0 O | 2A A oN | 0 AS 0 0 (39)

Qn fFy Sy\ oFy = ——| — + — —_ eee

now e,{ (Bea)(F -l/F(S-Bles «G

Equations (35), (37), and (38) may now be written as

A454 + B,4, + Ci Zsa = O (4 = 1, 2, eet, N - 1) (40)

Dozo + Eo = 0 (41)

Pyen-1 * PyAy = 0 (42) 34

For such a set of homogeneous equations, a recursion formula for 4; may be written as - (i =1, 2, +--+, N- 2)

and Gs Z, + P.Z,.. = 0 3 tot dd (1 =0 if Z #0) (l=N-1 if 2 #0) NM where P, isa (4 x 4) recursion matrix. To find P,, combine equation (43) and equation (40) to obtain

z, + (B, -~ A.P ae ) -1 CaZ 0 (i = 1, 2, «++, N= 1) (44) 1441 7

Comparison of equation (44) with equation (43) shows that

-1 P, = (3, - A,P,_,) Cy (i = 1, 2, «++, N- 1) (45)

Comparison of equation (41) with equation (43), the latter written at i = 0, shows that

P =D -E (46)

From equations (45) and (46), P, may be found at all points with the exception of the point i =N. This process of determining all required values of P, in terms of Py is in essence a Gaussian elimination process.

Equation (43) written at i =N- 1, in combination with equation

(42), yields (Py - ByPy-i) 2 = © (47) 5?

If Zany # O, then for a solution to exist,

[Dw ~ ByPya| = 0 (48)

Therefore, any frequency parameter af which satisfies equation (48) contains a natural frequency of the system. The natural frequencies can be found by trial and error by selecting successive values for a, calculating the matrices of equations (39), and using equations

(45) and (46) to evaluate the determinant in equation (48). This procedure is continued until the desired zeroes of the determinant are found.

The method must be slightly modified for the case aN = 0.

Substituting equation (43) written at N- 2 into equation (40) written at N-1 yields

(Byer - AveaPuee) Zen = ° (49)

If Z Nel = 0, then Z = 0 from equation (43), and the solution is 1 trivial. Therefore

[Byea > An-aPy-2| = ° (50)

Consequently, for the case Ay = O, equation (50) is used in place of equation (48) in the elimination process.

After the natural frequencies have been found, the corresponding mode shapes are determined from equations (43) and (47). Equation

(47) is used to find a normalized Zy: For the case where ay = 0, 36

equation (49) is used in place of (47) to solve for Zn 3" The remaining Z,'8 are then determined by using the recursion formula, equation (43). To find a normalized Za it is noted that the components of an are proportional to the cofactors of the elements of a dependent row of the coefficient matrix in equation (49). If

Oa is defined as the cofactor of the jth element in the kth row, where row k contains the coefficients of a linearly dependent equation, then the normalizing factor 5S can be defined as

5 = V@rx) + (Oo) + (85,)° + (Sux) (51) so that An may be given as ork Coie

il i (52) Ole a om 3k € Ohi

The index k indicates any row which is a linearly dependent row of the matrix (Dy - EyPyaa):

This numerical procedure is particularly well suited for use with a large number of stations since only the band elements need be retained during the computation process. References 2, 3, and 5 give further advantages in using this general method of solution.

Although equation (48) (or eq. (50)) contains all the roots of the system of equations (40), (41), and (42), this method of elimination introduces spurious singularities in the determinants of equation (48) or (50). For some shell configurations and boundary a{

conditions, it is found that the roots and singularities very nearly coincide, and the usual root searching methods fail to indicate a root if the increments given to the frequency parameter are too large.

Moreover, some of these singularities are associated with a change in sign in the value of the determinant even though no zero exists at that value of the frequency parameter. The technique used in this investigation for avoiding this difficulty is presented in Appendix B.

A set of linear equations governing the infinitesimal vibrations of axisymmetrically prestressed shells has been developed and both the in-plane inertia and prestress deformation effects have been retained in the analysis. The equations derived are consistent with first-order thin-shell theory and can be used to describe the behavior of shells with arbitrary meridional configuration having moderately small prestress rotation.

A numerical procedure has been given for solving the governing equations for the natural frequencies and associated mode shapes for a general shell of revolution with homogeneous boundary conditions.

The numerical procedure uses matrix methods in finite-difference form coupled with a Gaussian elimination to solve the governing eigenvalue problem.

Accurate results can be found for specific shell structures with this procedure. However, even with high-speed computers, it becomes impractical to attempt an extensive parameter study of the behavior of shell systems. To achieve a more rapid analysis of the effects 38

of the different parameters governing the dynamic behavior of shell systems, an approximate set of equations of motion is developed in the next section. The equations are limited to cylindrical like shells having a shallow meridional curvature. The finite difference solution of the general shell equations developed in this section can then be used to determine the accuracy of the approximate solutions. VIII. APPROXIMATE SHELL EQUATIONS

A. Derivation of Approximate Shell Equations

In this section, a set of linear homogeneous field equations is developed which governs the dynamic behavior of doubly curved shells with positive or negative constant shallow meridional curvature.

The nonlinear strain-displacement relations given by reference (10) are specialized for a cylinder and modified slightly. The cylinder is given a small axisymmetric deformation, then, by removing the strains caused by this initial geometric perturbation, an unstrained state is established in the deformed configuration. The nonlinear equilibrium equations corresponding to this unstrained state are developed and linearized.

The first approximation nonlinear strain-displacement relations developed in reference (10) based on the assumption of small strains and moderately small rotations become, for a cylinder with radius R,

« 2 ~ “~é = n~ tz1 “ 2 1 a 79 p= Ue ta (*,) + +3 4 -——s

n~ “ oO Vy “A u “a _ 9 W iL A a 1 A 26

£4 = z + R + ope (®59°- V + a (@,, - R |

1] 2 Yxg Woe 7, a fey = 5 V 7€ + R +t (4, - # —

a9 ho

Ro ~ = (#4 ~ ¥) 9

A t| (3 a0 Usy “ea = R (5, - o)e "TR (53) where the symbol “ is introduced to indicate quantities measured with respect to an unstressed cylindrical surface.

To permit the method of solution which follows, the term Reg is modified so that the term - 3 is eliminated. The quantity Reg may be written as ‘ (*s0) ®eo = - (he > *) 2 - aR 0%) where (® 0) contains only the linear terms of Egg. It is shown L € in reference (19) that linear terms of the type = multiplied by a numerical factor of the order of unity may be added to the linear expressions for bending strains without introducing an error greater than that originally introduced through application of the Kirchhoff-

Love hypothesis. Hence, K E9 is sufficiently defined by

Reg = (#5, - +) (55) lh lh

To increase the manageability of the strain-displacement relations, only the Von Karman type nonlinear terms are retained (i.e., nonlinear terms composed of products of derivatives of the normal Ay

displacement). The final form of the cylinder strain-displacement relations reduces to ,

A A lL osa 2 é, ge >= Ye += TB (¥ 6) 5 A 9 W 1 A 2

e = 2 + —- + — Ws 8 R R DRO ( 5}

~ _1lfa 9 la a»

(56) Keg ~ 7 Mogg

“A 1 nN “~ fg = 7 73 (Hog - ¥) Og R

”~ 1 NN Os Reg = - F(%-*) J

A set of strain-displacement relations governing the behavior of a shell of revolution with a shallow meridional curvature is developed in a fashion similar to that of references (11) and (12) by the introduction of an initial displacement wo(§) to the cylinder middie surface. Let

W(E,8,t) = w(E,8,t) + wi(§) (57) where w is the displacement measured from the initially deformed

surface along the normal to the cylindrical surface. Since the ho

meridional curve is assumed shallow, it is sufficient as a first

approximation to represent the initially deformed curve by N ur | CO WI wo(8) ~ DR o~ uve

where R, is the constant radius of curvature of the initially E deformed meridional curve. This representation restricts the

subsequent analysis to shells of the form shown in the sketches in

figure 43. The axial displacement u may be represented by

6-0 + He (59) uw where for a shallow meridional curve (i.e., R, large), u is the E meridional displacement measured along a tangent to the initially deformed surface. With application of equations (57), (58), and (59)

the strains caused by the total deformation can be written as

~ 2 ao = ”™ W 1 ¢~ 2 1 /fé )

“A ~ 2

@ = "6 + W + L “8 Bm 8°” R R eo R eR, R

u Wy. W A lfa 7g 7€ "78

Seg -3(%, + R + R (60) A ~ 1 K é = “ee _ ER——

“nto <= 4s1 (og~ - 9).gnw R “ 1 ~ “A “eg -- § (%, -¥) J 43

The strains caused by the initial deformation Wo are

2 E 40) £ 2. oS eR, R

€ £8 = 0 (61) Ko eek E Re

Ke _ 0

K £8| = QO

The initially deformed system is now taken to be in an unstrained

equilibrium state. Any deformation u, v, Ww away from this new

equilibrium state introduces strains given approximately by

€=ae. €,, thus, from equations (60) and (61)

~ > ° “Geta te (me)

~ ~ a ~ Vs 78 ,-w,1;~ 78 “e= -R F2(%)

~ze uily~ ye,u; W, Ws ewe gO 2 \7E OR R \ yy

Ke = 7 Woge (

Ry g = - =aRe (Wo 6 - ¥) ,9

~ --2(q - ¥) Keg = R Woo Vv st / (62) where & is now a length measure along the meridian and where the quantity v = ¥ is introduced for convenience in notation. Equations

(62) are the strain-displacement relations for a shell of revolution with a shallow meridional curvature. To obtain a set of equilibrium equations consistent with equations (62), the principle of minimum potential energy is applied.

The total potential energy for an isotropic, linearly elastic, homogeneous, vibrating shell acted on by a uniform lateral conservative loading p is

76 ~\2 Tl = oa E 2) Je,re 2 f fP™ P 7B n ( (nw ¢ +2 ~\2re | + (E, + Z Ro)

+ Ou (E + Z Fe] (Z, + Z %) + 2(1 - yu) Ge + Z Rye) paaaoa

fFe penn 1 D pen Pde Jo Faaoes = By 8 i , [Gs

+ (74) + fF Rdzdodeé (63) 45

Performing the integration over z and allowing the variation of the energy to vanish yields

r§5 pen ~ ~ we ~e Ne ~ M “ne oll “Uy J Ne Of + Noy SE, + Neg OF eg + M, Sk, + 9 Sk 4

~e ne ne ~ ~N ne “we ~ + Me g OF eg - pow -- vh (i, Suy, + Voy OV», + Woy si )} 9 dé = O

(64) where . W, = B(E_,+H &)

No = B (E, + e, |

Neg =B(1- 4) Eg (65)

Me = D (Fe + Rg) Mg = D (Fy + | Mpg = D (1-4) Keg J

and where B and D are defined in equation (16). Integration by parts in equation (64) leads to the following nonlinear equilibrium

equations 46

N ~ E9,9 ~ _ ° Ee + - vh Usiy = QO

N M OM 0,8 ~ 6,0 EO,E ~ ~

A * Neo, 7 2 * R - vhv,,, = 0 (66)

M + ag6, 66 + N8,68 - ne - a + N w + N we | E, EE R R D- - RORE e Wee * Neg R 9

1 (5 Woo ~~ nw ~ we + = Ny z + Neg Woe 5 +p-vh Wop = O ? a with the following prescribed at the € constant boundaries

Ne or u

OM, .

Neg + = or Vv \ (67) 2M w ~ £9,909 nN ~ ne 79 ~ Mee + 2 + Ne Woe + Neg RO or WwW

M, or Ws 7

These equations parallel equations (2) and (4) in Chapter VII. The

equations are linearized for prestressed linear vibrations by the

same procedure as that used in Chapter VII. Let 47

Ae = N, (8) +N _ €,9,t)

Ae = G9) + N,(§,9,t) 4 = M,(6) + M,(£,9,t)

me me = M)(&) + My(6,9,t)

Also let

= u(&) + u(£,9,t) ae

v = v(€,9,%) we

= w(t) + w(&,9,t) He ) where the barred terms are associated with the stresses and deformations due to an axisymmetric loading and the unbarred terms are associated with stresses and deformations due to a subsequent linear vibration about the prestressed state. The equations governing the linear vibration about the prestressed state are found by applying 48

equations (68) and (69) to equations (66), applying equilibrium of the prestressed state, and neglecting vibration terms. They are as follows:

N ~ Neve 0,9R - vh Use _= 0

N 6,8 M 6,0 OM E9,& R + Nove + —L— as: 2- - Vb Voi, = =O (70)

M + e888 + Ho,99 _ 78 _ 0 +N, wy,, + “e Ww g,&& R Re pRe " D_R € 7 > && “aae > 99

- vh Wei = 0 J

where the classical assumption Woe = QO has been made and where it is assumed that the prestress loading is such that Neg = 0.

The corresponding boundary conditions are

(71)

The underlined terms would be omitted for a Donnell type approximation.

The corresponding terms which would be omitted as a result of this

approximation are similarly underlined in all subsequent calculations.

If these terms were omitted, the system of equations would reduce to

those given in reference 11. Equations (70) and (71) parallel equations (13) and (14) in Chapter VII.

The vibratory stress and moment resultants are found from

equations (62), (65), (68), and (69). With the prestress deformations

neglected, the resultants become

W Vig W

= | B oe + Ey tu (2 += : utr utr

Vig .

9 { =| Ete (oetk), =

B(l - p) fe Neg = Haw) (6 * ve) (72) Me = - Df woes +3 (¥ +6 - Xo

My = - o| % (We - v) 6 + ou “ee

M Lo = = PU =u) (Ho -¥) J]

With the aid of equations (72), equations (70) are written in terms of displacements as 50

(Lae Wg 0° W999) (2- yA 4 R R,} R je 32 12 7 £E0 E R

vh B tt= ° 73)

ioyh u.,. t+ L,u\ 28 ey aM(2 =) Ry R|R ~? E R Re R ~ 12 n° ee” le EE 8

A 2 [s 2 1 Ou +——

N Ny w +o5\w-—w,1 E -—-—>~t+—v¥W,9 ~798 vh = 0 ts B gE 8B Ro B tt J) where now A= = . Similarily, the corresponding boundary conditions at €& = constant are

ne3 - - u) Viggo t Ro 2 Wage, + (2 - up) ¥, a4] Bowe,Ne =O OF w=

Wee tsH (6 - x) 57 = OQ or wy, = 0 R ’ (73d)

With prestress deformations neglected, the prestress stress resultants are defined by

N, = N E (74)

where N is an applied meridional stress resultant and jp is an applied constant lateral internal pressure. Equations (73) are constant coefficient partial differential equations. If prestress deformations had been retained, additional terms with variable coefficients would occur and approximate methods of solution similar to those used in Chapter VII would be required. The solution to the prestress equations are given in Appendix C.

B. Solution of Approximate Vibration Equations

Equations (73) are satisfied by a solution of the form

AX u-=zA. h, e J sin ng eit J od Ax jot V= A, g, e’ cos nde (75)

Asx w-=A.e J sin n9 eit De

where A, are the characteristic roots of equations (73) and the

nondimensional variable x = : has been introduced.

Application of the assumed solution to equations (73) yields

—— le > MA FFL HA, fs, hy A, 0 )

fd fod 4+ fon +f ge. A -lo\ 25 4%; 5 6°93 7 fy

2 4 2 fA35 f 6s +f, fa A, + fy A, * “10 A, 0) Ne” ~ NS - (76)

where »

- & 8” 12

fe no né _ N 9°” ~~ 6 "8B

a2 4 N Non ) 8 2 2 fio _ 3 + (KO + Qu +1) + _zn - 9 y (77) where » = ana 92 - O_#oO We v(1 - us)2

x Re EB

The characteristic equation is found by setting the determinant of the coefficient matrix of equation (76) to zero. This yields the biquartic 8 6 4 2 A; + ae A; + a), A, +a , ta =0 (78)

where

f,(2fnf. - f3f,)- 2 fitpt, “(£7 ° +f fofotty EE +f,f,- (f, "| fy 3 (79) f, (7Fafe - tty) - f, (Pf, +f 1°6) +f if5%g + fy s +f,f, +63) fy P56 f, fg

- (8) + fo I, +f, f, - 2] + fy fg

Ae Py tp 54

The amplitude coefficients hy and 8; may be found from any two of the equations (26). From the last two equations of equations

(26), they become )

ne - (#6 ry + t,) + (tg r," + £9 ry + £45) (ty re + £5] J tA, (t¢ ry + £.,] - fA, (fy, reo + f| | (80)

at f (fg r," + fy re + F) +f, (26 ro + £,) J fy (f6 ree + £.| - f, (A, reo + 5) a“

In general A, can be complex. Since the displacements u, v, and w are real then A,» Ay and g are complex whenever A, is complex.

Since the complex roots A, occur in complex conjugate pairs when complex roots do occur, it can be shown that the corresponding quantities Ay» Ass and B, will also occur in complex conjugate pairs. For axisymmetric vibrations (n = 0) the terms g vanish and the circumferential equilibrium equation uncouples from the remaining equations so that the procedure must be modified (see Appendix D).

The complete solution for u, v, and w is found by summing the contributions from each characteristic root thus li

c U(x) sin né eit

v = V(x) cos né eit (81)

w = W(x) sin nO ett D2

where

U(x)

V(x) UJ > (82) ba

ce” VW VW W(x) @ oP

[Jo jal

The stress resultants are found from equations (72) to be

Lot td n(x) sin nO e x

“Ol

Lwt

bd ny (x) sin n9 e Po

Lot bd ng (x) cos nd e

De m, (x) sin nd ett

m, (x) sin no ei”

m, (x) cos nO e Lwt &9

where 56

Ax ng =),= Ay j [B= - may . tuktH +uh,tH A,NS] le J j=

8 Lew Asx no > ), ( B ) AsCas + 8s Ay) e j=l (84) 8 5 Aj

B B aLB c (7 - 85) 7] « j=l -

[: 5 Ajx

= | A nme a)-w ry) « Oo £8 [P 8) 8% J=

8 A.X sa> ), O-8) 5 [(8 79) | ed

J= J

The boundary conditions may be found in terms of displacements from equations (71) with the use of equations (72) and with the solutions (81) and (82) may be written in general at two edges as

(fm +H] ®)-© (85)

where {ay is an 8 element column matrix whose elements are A. and where N, Y, ®, and are 8 x & matrices whose elements are given by A. S J 2 N.1j = ( 1 - ng 5).) tk x +h. prpeaA,

A. 8 n > N53 = nh, + 8; += (8) - 4) s}

A, S Je 2 =H nm, 2 3\ a Na, {8 (2- uw) n (85 - n) - ;| A, + TD A, e

r,s My, G (e; - 2) + | ef

A, So - ~_ toy = (u(t ~ ag) 4K +8 ry} e e

A. S no | -$— Nes = a +[eF @- 9], “

A. S

2 = 2 ._ J _( |AL N A 3 2 Nos {8 (2 - uw) n (8; - n\ - H A, + TD Aj | e 58

A. S eo o Co NV = n ~- niyta e 83 Gc }

r, S

yd Y.. =h,e 13 J A, 8 _J [email protected] 2 "94 8 5 A. S J e Y.. = 33 “ A. S J

Y,. =A, e 2 4 j A. S .~ J 2 Y_. = h.e Dd J A. 8 ~ Co Y-.63 ~= 53 e vA. 8 ~ 2 eo Y_. =e 73 A. S _ J Cc ¥g, =; & J (86) where S = = is the total nondimensional axial length of the shell, and

®, O for 144

Vi; =0 for 1¢j (87) Vig 72> 255

The elements V5 take on the value 1 or O according to the prescribed conditions. The 9%_. and ¥05 matrices are used to select bd the prescribed boundary conditions in a similar fashion as in Chapter

VII with the a and BB matrices. If a stress-free condition is desired, the corresponding ° 44 term is set equal to one. If a displacement variable or meridional slope is constrained such that it must vanish at the boundary, the corresponding P55 term is set equal to zero. The ® and W matrices may be generalized to enforce linear elastic or directional constraints.

For a particular set of homogeneous boundary conditions and a particular circumferential harmonic mode number (n), the natural frequencies of the system are those contained in the frequency parameter 82 which cause the determinant of the coefficient matrix in equation (85) to vanish, that is,

jon + wy] = 0 (88)

Due to the highly transcendental character of equation (88), a trial and error procedure must be used to find the natural frequencies.

The procedure is exact in the sense that the frequency parameter can be found to any desired degree of accuracy.

A trial value for & is selected and the quantities in equations

(77) and (79) are computed. The characteristic roots are determined from equation (78) and h, and g, are found from equations (80).

The values for ®., which give the desired boundary conditions are 60

selected and the determinant of equation (88) is evaluated using equations (86). The value of the determinant is then compared with

zero. The frequency parameter is then increased by fixed increments until the determinant of equation (88) changes sign, indicating the presence of a root. The root is located to within the accuracy desired by successive halving of increments.

The residual in equation (88) is in general a complex number, but since the characteristic roots occur in complex conjugate pairs, the determinant must have complex conjugate columns and thus its value must be either real or pure imaginary. This facilitates the automatic

search procedures in the root searching operation since only one number must be corrected to zero.

The success of this method as a rapid means of obtaining

solutions hinges on the investigator's ability to make good initial estimates of the frequency. To aid in the selection of an initial estimate, a closed form solution is obtained in the next section for

the vibration frequencies of a shallow doubly curved shell with freely supported edges. The solutions to this system may be used

to select initial trial frequencies for systems with other boundary

conditions.

C. Solution of Approximate Equations for a Freely Supported Shell

A direct solution to equations (73) is available for the freely

supported boundary conditions (Ne =Vewe=M, = 0). Assume a E solution of the form 61

mr iwt u= U cos —— sin n@ e fs 5

v == V,, sinin Mtmmx=> cos n@ 8 e iwmt (89)8

wi W sin max sin n8 eet fs 5

where m is the number of meridional half waves. This solution

satisfies the freely supported conditions at x =0O and x= 6.

Equations (89) are substituted into equations (73) to yield a set of linear homogeneous equations. For a nontrivial solution to exist, the determinant of the coefficient matrix of the resultant set of equations must vanish. This procedure leads to the characteristic equation -0 +5, 0-5, oP + -0 (90) where the coefficients A, are given in Appendix A. Hence the natural frequencies of a freely supported doubly curved shell of revolution with shallow meridional curvature can be found for any

specific mode by solving equation (90). This equation leads to

three frequencies for each set of m,n considered since in-plane

inertia terms have been retained.

D. Membrane Solution of Approximate Vibration Equations

It is of value to inspect the extreme case of zero bending

stiffness (D = 0}. The vibratory behavior in this case is associated 62

with only the extensional properties of the shell. The membrane equations are found directly from equations (70) by deleting all moment terms, yielding

N £6,6 Nee + +R =O

N 8,8 + = ‘1

N N w é 8 — = 78 _ R + zr Ne Woee - Nog SS + yh Woot = 0

where for convenience in this limiting case the in-plane inertias have been neglected in the first two equations. The first two equilibrium equations are identically satisfied by the introduction of the stress function wt defined by

) Ne— =—Vs 96 > =U,u.. +¥— ru(se+¥)—_— a B Ro E Re R R

N Vv 8 76 W Ww

Bo RO (4) Yee 7 R J

in terms of the nondimensional variable x, the third equilibrium equation becomes

KYSE ¥eg9 RTVox BaNy. 7 BeeNo FBvh Re ge =O (99)

It can be seen from equation (93) that for a negative Gaussian

curvature (kK negative) unstressed membrane shell, the governing equation has a hyperbolic character. The equation for the cylinder has a parabolic character and for the positive Gaussian curvature shell has an elliptic character.

1. Solution of membrane vibration equations for a freely supported shell.- Equation (93) is satisfied by

y = Vi, sin.. —5=MAX cos n@ e Lot

(94)

w-=W_ sin — cos n@ ert m S where it follows from (92) that

a= U_ecos mix cos ng oot m S (95)

v= V_ sin mx sin n8 eit m S

This solution satisfies the freely supported boundary conditions.

With the application of equations (94) and (95) to equations

(92), equations (92) may be written matrix form as an ( | [75 mx un k et + U tn ~ ne | ym 1 + uk 4 (88) (96) po BST n WAX m " R “VS \ | ra en (=)mr -n — O W tl -u 8 J hn ae 7 men

a Vv =< (97) men R (4 _ 2 (2) . 2 ,

Substitution of equations (94), (95), and (97) into equation (93) yields the following membrane frequency equation 2 2 (a - u®)2 (x, + 8°) 2 =Hy =ER a\ 2 (2 ) = + — Bp n (98) mem, (1+9?) D 2

mr where 8B = ns is the ratio of the circumferential to the axial wavelength of the vibration mode. The frequency determined from equation (98) vanishes for an initially unstressed membrane for neridional curvatures given by

k= - 8 (99)

Thus, for certain mode shapes the vibration of a negative curvature membrane shell with freely supported edges is not sustained by the membrane stiffness of the shell. 65

2. Characteristic roots from membrane theory.- The membrane

characteristic equation is found from the exponential solution forms

x NXe > y = V5 ed cos no eet

+H AX2 we W, e J cos ng etdt , (100)

x 2X u= U; e J cos nég elt

% Ax. v= V5 e J sin n@ eit J

Substitution of this solution into equations (92) yields 2 7 x vi (a,° - n°) Gon) y (a - a") (x, no my

Substitution of equations (100) and (101) into equations (93) with prestress terms deleted yields the biquadratic characteristic equation

c - a) - | r;" + One [a - ky, @ - 2)| ry

ent [G2 -u8) a - = 0 (102)

This equation may also be found from equation (78) by deleting in- plane inertia terms and products of rn in the functions given by 66

equations (77). Solution of equation (102), yields the following membrane characteristic roots

2 = —*n—s |, (. -u°) - 0% 20 (0 -x,) he J membrane (a - un) (103)

E. Pure Bending Solution of Approximate Vibration Equations

The extreme case of pure bending is considered in this section.

The vibratory behavior in this case is purely flexural and the middle surface extensional strains are assumed negligible. By assuming that the extensional stiffness B remains finite, the pure bending equations are found directly from equations (70) by deleting all vibratory stress resultants. The displacement formulation may be written down directly from equations (73a) by retaining all terms containing rn“, v and w inertia terms, and membrane prestress terms. The following equations result:

2 2 AN 1 -u\ .2 AY (2 -u) ,2 i2 799 * (ee) A Yoxx 7 TS W999 7 qa Wo xx 2 2 _ VR Vig } v, = 0 E tt » (204) a M(2-ud) Ng 12 "7600 12 xx@ ~ 7D V7 BO Wexx 7 BW g9

2 2 + VR (4 - = 0 67

The meridional displacement u and the meridional curvature KE do not appear in equations (104), thus in the approximate formulation the pure bending state is independent of u and Ke

The freely supported boundary conditions are satisfied by

mmx iw be w=W b sin —s— sin n8 e Pp B (105) iw. t v= V sin art cos nO e pb pb 5

If the Donnell approximation was made, the v inertia term would vanish and the natural frequency of pure bending would be equivalent to the natural frequency of a simply supported rectangular plate and would be given by

2 2 N N =a5_N ( (8 2 +1) } a 4 +( SBx ,2 +g 6) jn 2 n #0

()Pp Donneil bh

(106)

Substitution of equations (105) into equations (104) yields the following pure bending frequency equation

2 _ 2 7+ 8 820 a, 11 *ta,,\e #22 5 2 Q pb = a et = —___—2 ) ~ an B20 -~ (2) | (107) where ~

o> Q4=5 E + 2(1 +u) B 2 | n 2

holt

Pr a= 7 ts l2+ (2-H) 2°] n? ) (108)

2 Sop = es Donnell J and where again

Equation (107) gives two pure bending frequencies one of which

is close to zero. This frequency is associated with a predominant

in-plane v displacement mode and is usually several orders of magnitude higher than the frequency associated with a w predominant mode. These frequencies being associated primarily with in-plane motion are governed almost entirely by membrane action, thus, the pure bending theory cannot give a reasonable estimate of these frequencies since its stiffness contribution is negligible, so that frequencies close to zero are expected and can be ignored. Further- more for n> 3 the pure bending frequencies found from Donnell theory which are associated with the w inertia term only (equation

(106)) will closely approximate those found from equation (107). IX. RESULTS AND DISCUSSION

Vibration calculations have been made for some simple shell of revolution configurations to determine the accuracy of the approximate analysis developed in this paper and to determine the influence of meridional curvature on vibration modes and frequencies. The accuracy is assessed by comparing the results of the approximate analysis with results obtained from the more accurate numerical analysis presented in Chapter VII. The influence of meridional curvature is assessed by performing a parameter study for shells with differing meridional curvatures as the length-radius and thickness-radius ratios are held constant. Results of additional calculations based on membrane theory and pure bending theory clarify the relative roles of membrane action and bending action and their dependence on modal wavelength ratios during the vibration of these shells. The results obtained are for a specifie class of shell configurations; however, the effects determined many provide insight into the general behavior of more complex doubly~-curved shells of revolution.

A. Shell Configurations Investigated

The class of shells of revolution with constant shallow meridional curvature was elected for investigation since it has the simplest configurations for fulfilling the parameter study of interest. The equations defining these shells are given in Figure 3 and are used whenever the more accurate numerical method of Chapter VII is

69 70

Positive Gaussian Curvature Shel] R70

\y k= Rp

‘I

+ =

—_

4 Zero Gaussian Curvature Shell (Cylinder)

s=SR kK = 0 1 r =] l _ } kK, = 1

Negative Gaussian Curvature Shell Re < 0

Equations same as positive shell with

R. € taken as a negative number

rot

Figure 3.- Geometry of class of shells investigated.

employed. Since the approximate theory is limited by shallowness in the meridional direction, it is sufficient to represent the meridional curve of this class of shells by equation (58) when using the approximate theory (see reference 20). The effect of curvature is determined for both the approximate solutions presented here and the more accurate numerical method of Chapter VII by varying k while holding the length-radius and thickness-radius ratios constant. The mass distribution varies slightly as KL is changed, but if meridional shallowness is maintained, the effect of change in mass distribution on the natural frequencies is negligible, and any change in natural frequencies can be attributed solely to the change in Gaussian curvature of the shell and the resultant shell stiffness change.

When the numerical procedure of Chapter VII is employed, 200 equally spaced intervals along the meridian are used. In all calculations, Poisson's ratio is taken to be equal to 0.3 (i.e. wu = 0.3).

The central rise of the shell meridian (c in figure 3) is given by

c= Re (2 - cos Zr, | (109) where s is the total meridional length. The shell meridian can be approximated by the first term of a series expansion of the right- hand side of equation (109). ‘The percent ratio of the central meridional rise to length can thus be represented by Sk T= = (100) (110) (2

This quantity is introduced as an auxiliary parameter which is used

to give an indication of the degree of shallowness of the meridian.

B. Accuracy of Solutions

Before a parametric study is performed using the approximate

theory, a level of confidence in the accuracy of its solutions must be

established. To make an assessment of the errors involved when using

the approximate theory, the results obtained with this theory are com-

pared with corresponding results obtained with the more accurate

numerical method of Chapter VII for several specific doubly curved

unstressed shells in figures 4, 5, and 6. In these figures, the lowest

frequencies are plotted for successive circumferential mode numbers for both positive and negative Gaussian curvature shells. The cylinder

results, found using the approximate method, are repeated on each

figure for purposes of comparison. Since the only approximation made

for the cylinder results was the neglect of small nonlinear terms before linearization, only terms related to prestress would be

effected; thus, the unstressed cylinder results presented in these

figures are exact.

For the slightly curved shells (tr = #1.87) of figure 4, the approximate theory is shown to be quite accurate with a larger percent-

age error evident for the negative Gaussian curvature shells. The

accuracy diminishes as the curvature of the shell meridian increases as is shown in figures 5 (Tt = 3.75) and 6 (Tt = £5.62). The positive

curvature shell results never differ from the more accurate results by more than 8 percent. The negative curvature shell results have larger

errors but still indicate the general character of the actual solution. 13

1.0

S=3 # =1000 o numerical solution k,=+. 05 ot o numerical solution k=— 05

: e— approximate solution a Cc wo => D i i) 2c c ® = ec S 01 k, = +.05 5 & "\, & ™—~ = oe —_——— ‘os cylinder .

‘o ‘N

"S xX ‘ 7Je \ m=2 k= — 0 ye a

m=]

al Joy ft tNve py ty

a if 2 4 \J 6 8 10 12

o Circumferential Mode Number, n

Figure 4.- Comparison of minimum frequencies from approximate analysis with those from numerical analysis for freely supported shells (x, = + 0.05; S = 35 &R = 1000). 7

1.0

S=3 ’ RL

a o numerical solution ky=+.1 S o numerical solution k,=-.1 =. —— + approximate— solution = rc S 2 ky = +1

ws S OeGa = 01 E cylinder 2 oo

£ = m=

e o

® . & oF ? eo ° °o eee OF 5 27 m= ”

Cm, -° o”

. A n=?

on LL tt | | | | | | t |] | J 6 8

Circumferential Mode Number , n

Figure 5.- Comparison of minimum frequencies from approximate analysis with those from numerical analysis for freely supported shells R (%, =+0.1; S = 3; i 1000). 19

1.0

S=3 R _ _ 1000

so numerical solution k. =+.15

o numerical solution k=—-15 2

—~ «— approximate solution Frequency, Frequency,

0.1

Nondimensional Nondimensional vo

/ N Minimum Minimum ' ° 4

\ Leoo 4 7 “en, . a” g \ nS

gph _-L_—_1 1 tb tb tt

0 2 4 6 8 10 12

Circumferential Mode Number, n

Figure 6.- Comparison of minimum frequencies from approximate analysis with those from numerical analysis for freely supported shells R (x, = % 0.15; 8 = 3; a= 1000) . 76

The lowest natural frequency for the axisymmetric vibration mode n=0O is associated with a pure torsional mode and is independent of the meridional curvature in the approximate theory. The more accurate results, however, indicate that this frequency is dependent on curvature

40 a small degree as can be seen by inspection of figures 4, 5, and 6.

The approximate procedure has a modal solution of sinusoidal form

along the meridian (see equation (89)) whereas the mode shape in the

numerical procedure is calculated once the frequency is determined and

need not necessarily be sinusoidal. Plots of the minimum frequency

normal displacement mode (w) determined using the numerical procedure

are given in figure 7 for particular values of n for the negative curvature shells of figures 4, 5, and 6. In each case, the number of

axial half-waves (m) determined by the approximate theory agrees with

the modes given in figure 7. As the curvature increases; the mode

shape begins to deviate from sinusoidal form, thus the sinusoidal

modal solution of the approximate theory becomes a less accurate

representation of the true modal configuration.

A comparison of the results found by the approximate theory and

the numerical procedure for a long negative Gaussian curvature shell with a rise-length ratio greater than 6 percent is given in figure 8.

Large percentage errors are evident, with the approximate theory in

general overestimating the lowest natural frequencies. The

corresponding normal displacement modes given in figure 9 are found

using the numerical procedure and exhibit noticeable deviations

from the assumed sinusoidal form of the approximate solution

especially as n increases. Nevertheless the approximate solution

Displacement

Meridional Distance

‘Figure 7.- Comparison of w meridional modes for several n for successive degrees of curvature, (*, = -0.Q01, -O.1, -0.15; s = 33 ¢ = 1000).

(All calculations based on numerical analysis) 0.1 ae ae Ys \\ ~ \\ SW"

G Xt o Cc o = s in % Cc 2 c 0.01 £ Cc 2c & ~ £ £ = $= 10

f=R 1000 KR= RW TT

*-—— approximate solution

° numerical solution

jt | [| [| jf Jj [| | Jf J 0.001 2 4 6 8 10 12

Circumferential Mode Number, n

Figure 8.- Comparison of minimum frequencies from approximate analysis with those from numerical analysis for freely supported shells (, = 40.05, S = 10; 5R = 1000}.

2 —

COC

ME n=9

n=l0 n=l

Meridional Distance

Figure 9.- Meridional modes of w for several successive n R (&, = -0.05; o = 10; B = 1000}. (All calculations based on numerical analysis) 80

still exhibits the general shell behavior with respect to both the modes and frequencies. The radial displacements are larger near the ends of the shell as these areas behave as regions of low stiffness due to the larger crossectional radius. Since the approximate theory can only admit a sinusiodal solution, it cannot reflect this property of the shell. In positive Gaussian curvature shells, the region of low stiffness is in the center of the shell meridian and the modes calculated by numerical procedure (not shown) would generally yield larger radial displacements in this area.

Based on the comparisons made in this section, it is believed that the approximate theory gives reasonable estimates of frequencies and predicts trends adequately for values of T between +5 percent.

All analyses in the remainder of this report will be limited to shells contained within this range.

C. Effects of Meridional Curvature

The natural frequencies of a shell structure are closely related to the effective stiffness of the structure in the sense that as the effective stiffness increases, the fundamental frequency increases.

The positive curvature shells exhibit a strong stiffening character with the lowest natural frequencies increasing as the curvature of the meridian increases. For example, the fundamental frequency of the positive Gaussian curvature shell in figure 4 is 150 percent higher than that of the cylinder, while in figure 6, the lowest natural frequency is over 300 percent higher. On the other hand, negative 81

curvature shells exhibit rapid losses in effective stiffness and the minimum frequencies seem to be highly dependent on the circumferential mode numbers.

The lowest frequencies for each circumferential mode number always occur at the simplest meridional mode m=1 for positive and

zero (cylinders) Gaussian curvature shells. However, the negative

curvature shells exhibit lowest frequencies at higher meridional modes in the higher circumferential mode number range. This is well demonstrated in figure & where the fundamental frequency occurs for m= 2 and higher meridional modes are associated with lowest frequencies for n> 3. This figure shows an interesting phenomenon in the spacing of the frequencies. As n increases, the spacing between frequencies associated with successive meridional mode numbers

decreases. This behavior suggests that experimental resolution of

individual natural modes would be difficult to achieve at the higher

n range.

The results of figures 4, 5, and 6 do not show a well defined

trend in the behavior of negative Gaussian curvature shells, thus a more extensive parameter study is necessary. The fundamental

frequencies and modes of a series of freely supported shells with a

length-radius ratio of three (Ss = 3) and with the rise-length percent

ratio varying from -5 percent to +5 percent are presented in figure

10. The fundamental frequencies for the shells of figures 4 and 5 are

located by the vertical dashed lines. The horizontal dashed lines will be discussed later. As the positive curvature increases, the 82 | nN Ww | Aa WN en A = 1000 n

Frequency,

Perey Perey oq oq

Nondimensional

Minimum 01 001

Percent Meridional Rise

Figure 10.- Effect of meridional curvature on the fundamental frequencies of freely supported unstressed shells (7 = -5 percent to +5 percent; S = 3, a = 1000).

The meridional mode m=1 unless otherwise noted, the circumferential mode n is given on curves. (A111 calculations based on approximate theory.) 83

fundamental frequency and circumferential mode number increase monotonically. As the negative curvature increases from zero, the circumferential mode numbers decrease and each branch associated with a particular modal configuration has a distinct minimum. That is, for each branch of the envelope of fundamental frequencies in the negative curvature range, there are specific values of curvature at

Which a large decrease in effective stiffness occurs.

The fundamental frequency of cylinders and positive curvature shells occur for m= 1. However, an m= ce meridional mode is associated with the fundamental frequency for shells within a small range of negative curvature.

Figure 11 is a compilation of nine plots of the same type as figure 10. The length-radius ratio ranges from 1 to 10 and the radius- thickness ratio ranges from 100 to 1000. The minimum frequencies in general decrease as the length increases and as the thickness decreases

The reductions in stiffness in the negative Gaussian curvature range are more prominent for the thinner shells, but the effect is noticeable to some degree for all the shells. The minimum frequency for each branch of the envelope in the negative curvature range occurs at the same rise-length percent ratio for a given length regardless of the thickness of the shell. This suggests that the curvatures at which the minimums occur are related to membrane action. The degree of decrease in effective stiffness on the other hand is highly dependent on the thickness, thus bending action must be a prominent factor in the effective stiffness of the negative curvature shell in the region of these minimum frequencies.

84 tt] tt]

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, TTT) TTT)

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TTI TTI 3 2 .O1

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TUTTI TUTTI f f | , -O0| . |

Minimum

retry retry TTT TTT ~ D wn tT ; Rg

<< ph 100 43

no Ol

TTI

“TOP : : ; Lo ba a O01 | on wn

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Percent Meridionol Rise

Figure 11.- Effect of meridional curvature on the fundamental frequencies of freely supported unstressed shells for various lengths and thicknesses (« = ~5 percent to +5 percent; 5 = 1, 5, 10; = = 100, 200, 1000).

The meridional mode m=1 unless otherwise noted, the circumferential mode n is given on curves. (All calculations based on approximate theory.)

D. Membrane and Pure Bending Analysis

The results of the previous section imply that membrane behavior is closely related to the large reductions in effective stiffness observed for negative Gaussian curvature shells. The membrane equations which correspond to these shells have been solved in closed form for freely supported edge conditions. The solution for membrane natural frequencies is given in equation (98) in terms of the nondimensional meridional curvature (Ky ) » modal wavelength ratio (B), circumferential mode number (n), and prestress quantities. This solution is plotted in figure 12 for particular wavelength ratios for unstressed shells. The membrane frequency is a continuous linear function of KL for a given wavelength ratio and decreases to zero as KY approaches the negative value given by equation (99). Since the total mass and mass distribution are essentially constant as kL is varied, this decrease in frequency must correspond to a decrease in effective membrane stiffness. Therefore, for a given modal wavelength ratio, there exists a negative Gaussian curvature membrane shell with a nondimensional curvature k given by equation (99) which vibrates without developing any effective membrane stiffness (or for that matter any membrane stresses). As the meridional curvature increases negatively from this critical value, the membrane shell regains its stiffening characteristics.

To show how this membrane behavior is related to the previous results noted in figures 10 and 11, the modes (m = 1, n = 3) and

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(m = 1, n = 4) are investigated for shells with a length-radius ratio of 4 and a radius-thickness ratio of 1000 (shells of figure 10).

The wavelength ratios for these modes are 6B = 0.349 and 8B = 0.209, respectively. From equation (99), the meridional curvatures at which no membrane action is developed are k, = - 0.1218 and k, = -0.0685, respectively, which for the length-radius ratio in question are equivalent to the percent meridional rise ratios T = -4.57 percent and T = ~2.57 percent, respectively. These rise ratios very nearly coincide with the rise ratios locating the minimum frequencies in figure 10 at the modes in question. Hence, these reductions in fundamental frequency occur at curvatures for which the membrane theory predicts no membrane action. This can be shown to be true for the remaining minimums in figure 10 and all the minimums in figure ll.

The higher fundamental frequencies on either side of these minimums must be due to the reinstatement of membrane stiffness for that wavelength ratio as the curvature changes as indicated in figure 12.

The preceding suggests that membrane action is not developed in negative Gaussian curvature shells for certain wavelength ratios and that this accounts for the radical reductions in frequency observed in figures 10 and 11. This would further suggest that at these critical combinations of wavelength ratio and meridional curvature only pure bending action exists. If this be true then the fundamental frequencies found with a pure bending theory should yield frequencies near the observed minimums. To verify this, the pure bending theory 88

was solved. The pure bending solution given in equation (107) is independent of curvature and is in general a poor representation of shell frequencies since membrane stiffness is usually a quite prominent mode of resistance. The pure bending frequencies associated with predominant w displacement for a shell of length-radius ratio of three, radius-thickness ratio of 1000 and modes m=1, n = 3; m=l,n=4;n=1, n=5;m=1, n=6; m=2,n=7 are plotted in figure 10 as horizontal lines at appropriate places for comparison with general shell results. In each case, even though the pure bending results are independent of meridional curvature, the pure bending frequency closely approximates the minimum frequencies obtained using the approximate theory for corresponding m, n modes. Hence, it may be concluded that the reductions in frequency observed for certain negative curvature shells are due to a loss in membrane action and that for these critical combinations of modal wavelength and meridional curvature the shell is very nearly vibrating in a pure bending mode. Furthermore these combinations may be predicted from a simple membrane equation (equation (99)).

Further insight into the relative role of membrane action and bending action in the vibration behavior of doubly curved shells may be gained by inspecting the characteristic roots of the governing equations. Table I contains a tabulation of a sample of the characteristic roots found from the characteristic equation (78) for a particular circumferential harmonic mode number as the meridional 89

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curvature is varied from positive to negative values with particular

values of frequency maintained as constant. The roots are a function

of the governing field equations only and are thus independent of shell

length and edge boundary conditions. The character of the equations

changes radically with the sign of the curvature. A comparison with

the roots of the corresponding indicial equation from membrane theory

given by equation (103) shows that this change in the characteristic

behavior of the shell is associated with membrane (i.e. in-plane)

rather than bending terms. Moreover, the membrane roots are very

nearly unchanged by the presence of bending terms and in the shell

theory roots associated with bending terms are very nearly insensitive

to changes in curvature. Thus, in general, differences in the dynamic

behavior between negative and positive curvature shells are associated

with membrane behavior.

It should be noted that every negative Gaussian curvature shell will vibrate in a pure bending mode for some particular wavelength

ratio, however; if the m or n number its large, there may be

enough bending stiffness present to maintain a frequency well above

the fundamental frequency of the shell.

E. Prestressing Effects of Lateral Pressure

Figure 13 indicates the vibration behavior of a shell subjected

to a small internal and small external constant directional lateral

pressure. The prestress deformation is assumed negligible so that

the circumferential stress is constant and the governing equations 91

1.0 —

L. - S =3 a R L > 200

Ca 6 0,] ua > E

Ss . = ' 5 wm tc - = L 5 No r 2 an) =5 L 2x10 5 3 =z = 0 = = 01 — 5 = F 2x 10

oo. [ot yg Jf J ft 4

. -5 ~4 3-3 =-2 =] 0 1 2 3 4 5

Percent Meridional Rise

Figure 13.~ Effect of meridional curvature on the fundamental frequen- dies of freely supported shells subjected to a constant circumferential tensile and compressive prestress (7 = -5 percent to +5 percent; 5S = 3, é = 200).

The meridional mode m=i1, the circumferential mode n is given on curves. (ALL calculations based on approximate theory.) 92

retain their constant coefficient character. The internal pressure develops a constant positive stress in the shell wall which introduces a stiffening effect. This effect is more prominent in the negative curvature range causing the fundamental frequencies to increase slightly. The same small level of negative circumferential stress produced by an applied external pressure causes a radical deviation from the unstressed behavior in the negative curvature range with complete loss in stiffness (buckling) occuring for shells with a percent meridional rise in the vicinity of -4.7 percent. Only a slight decrease in frequency is evident for the positive curvature shells. This demonstrates the highly unstable character of the negative curvature shell in the vicinity of curvatures at which membrane resistance is ineffective and bending stiffness is small.

The introduction of large internal lateral pressure is a possible method of stabilizing the negative curvature shells. Figure 14 shows the effects of the application of a large internal pressure on the minimum frequencies of doubly curved shells. The fundamental frequencies are raised on the order of 1000 percent in the negative curvature shell so that the large reductions in stiffness at critical curvatures are no longer evident. The envelope of fundamental frequencies has flattened considerably with the result that only a small difference exists between the stiffness characteristics of negative and positive curvature shells. ‘93

1,0 = -

[ $=3

= Reh 1000

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- }-

__ | | _I 1 | 4 | 4 001

Percent Meridional Rise

Figure 14.- Pressure stabilization of freely supported shells G = ) R percent to +5 percent; S = 3; A= 1000).

The meridional mode m= 1 unless otherwise noted, the circun- ferential mode n is given on curves. (All calculations based on approximate theory.) Oy

The general result of varying the lateral pressure may be determined from figure 15. This figure is a plot of the fundamental frequencies for a series of shells as the circumferential stress varies. The fundamental frequency occured for m=i1 for all shells inspected in this investigation. The rate of reduction in stiffness as the compressive stress is increased is larger than the corresponding rate of increase in stiffness as the tensile stress is increased. The intersection of a curve with the abscissa yields the compressive buckling stress. The curves show that the negative curvature shells are considerably more susceptible to buckling at low compressive stresses than are the positive curvature shells.

F. Effects of Edge Restraint

The effect of edge restraint on the vibration frequencies of specific positive and negative curvature shells are shown in figure 16 and in Table II. The figure shows an increase in minimum frequencies of clamped shells (a =Ve=ewWwWe= Y x = 0) over those of freely supported shells (N,. =V=W=m = 0) for all curvatures examined. The boundary effects had a sizeable influence for moderate values of n but diminished in influence for high and low values of n. The clamped positive and zero curvature shells have an m=i1 type mode at the lowest frequency for each n whereas the negative curvature shells exhibit more complicated modes at higher n. Since in the approximate theory the lowest frequency for the axisymmetric mode

(n = 0) is the torsional frequency which has been shown to be 95

13 [~ +3% Rise 5

Al +2% Rise 10 09 +1% Rise

08 Frequency Frequency

.06 Nondimensional Nondimensional

.05 Minimum Minimum 04 Rise

N Nondimensional Circumferential Stress =z 10*

Figure 15. Effect of constant circumferential prestress on the fundamental frequencies of freely supported shells (7 = -45 percent to N - - +34 percent; 5 = 3; f= 200; = = -2 x 10 + to +2 x 10 +),

The meridional mode m=1, the circumferential mode n is given on curves. (All calculations based on approximate theory.) 96

1.0 —— r

——«-——= freely supported ———o— = clamped

S kK =+,] ~ a X > eo cS ‘\ S . Li 3 a 2 . “Stic ae © eg cree 5 eerreee bs wn c 0.1 beers @ 5 — kK =—] s = ‘ “=o 2° ° = = . TT ONw ae ‘“, ~ _ € _ , OS mo eRe s - m=4 = cylinder S . ‘ = 8

OL it ft | J] Jj [| J | | Jj | J 0 2 4 6 8 10 12

Circumferential Mode Number, n

Figure 16.- Effect of edge restraint on the natural vibration of unstressed shells (x, = 40.1; s = 3; - = 1000).

(All calculations based on approximate theory) 97

O= (ream) §4

JO 6190° 9G170° 9¢40° +2,470" L2ZS0° 940° T960° O¢O0t°O 9G¢e° O94T’ =

SHIONENOGMA A= (ooot

TVEAIVN

O=- (pea = 5 fodans *W=AMA=A=N

AHL {6 620" 9¢t0° 6GHT* 9G40° 970° ae ZQHT" Gace" 0960° 9zZG0° +),470° @1,90° =

NO ky 8 0 duyzs

INIVELSHY “Gor Gi ) =

aya]! (JUTBIYSeL x)

ADGH o= ou TISHS Mana

AO W374 = Q17600°

LOMITA 69T0° HEGE°O 90Z0" T9Z0" 2GcO" ¢620° 90¢0° 1L160° THNLVAUO 2 pedweto) TBuoTpTiom a=

FHL iy

FO TALLVOAN

NOSTYVdWOO

O= (peacoadns V *W=h=A= L6300° HEGE°O SQHT’ LLGO° g9TO" g0z0" T6Z0° Gtc0" cO¢0* €G20°

-°II AT8245)

WIVi “*N “*N

OT W T g i 9 ¢ G L 6 Q

independent of u, w and the meridional curvature Ke the minimum

frequency curves have the same value at n=O regardless of the boundary conditions (as long as v =O is maintained on each edge) and curvature.

The clamped negative curvature shells do not have the large reductions in minimum frequency which occured in the freely supported

shells. The results for different combinations of edge restraint of u- and Woy given in Table II show that the condition u=0O and not Woy = O caused the increase in frequencies above those of the freely supported shells. Thus, the membrane constraint condition u=0O has apparently prevented a loss in membrane stiffness for the negative curvature shells. The slope restraint Wy, = O was essentially ineffectual in its ability to raise the fundamental frequencies. Results similar to these have been given in reference

13 for circular cylindrical shells. X. CONCLUDING REMARKS

A set of linear equations governing the infinitesimal vibrations

of axisymmetrically prestressed shells is developed from Sander's

nonlinear shell theory and both in-plane inertia and prestress

deformation effects are retained in the development. The equations

derived are consistent with first-order thin-shell theory and can be

used to describe the behavior of shells with arbitrary meridional

configuration having moderately small prestress rotations.

A numerical procedure is given for solving the governing

equations for the natural frequencies and associated mode shapes for

a general shell of revolution with homogeneous boundary conditions.

The numerical procedure uses matrix methods in finite-difference form

coupled with a Gaussian elimination to solve the governing eigenvalue

problem. The solutions obtained by this method are used to determine

the accuracy of the approximate solutions used in the vibration analysis.

An approximate set of governing equations of motion with constant

coefficients which are based on shallowness of the meridian are

developed as an alternate more rapid method of solution and are solved

in an exact manner for all boundary conditions. The membrane and pure bending equations which correspond to this approximate set of

equations are solved for a specific boundary condition. The character

of the characteristic roots of these membrane equations are also

inspected.

99 100

The effect of the meridional curvature on the fundamental

frequencies of a class of cylindrical-like shells with shallow meri-

dional curvature is investigated. The positive Gaussian curvature

shells have fundamental frequencies well above those of corresponding

cylindrical shells. The fundamental frequencies of the negative

Gaussian curvature shells generally are below those of the correspond-

ing cylinders and evidence wide variations in value with large

reductions in magnitude occuring at certain critical curvatures. The

corresponding membrane and pure bending equations are also solved for

the same edge conditions. Comparison of the membrane, pure bending and

complete shell analyses shows that these critical curvatures represent

configurations at which the fundamental mode of vibration of the shell is

in a state close to pure bending. The membrane theory affords a simple method of determining the modal wavelength ratio at which the pure bending state exists for a given negative Gaussian curvature shell, while the pure bending theory gives a good estimate of the magnitude

of the frequency for this wavelength ratio. Meridional edge restraints

and internal lateral pressure reduce the wide variation of the

natural frequencies in the negative curvature shells and in general

raise the natural frequencies. External lateral pressure accentuates

the reduction in natural frequencies of the negative curvature shells and causes instability at low compressive stress levels. XI. REFERENCES

Kalnins, A.: Dynamic Problems of Elastic Shells. Appl. Mech.

Reviews, vol. 18, no. 11, Nov. 1965, pp. 867-872.

Budiansky, Bernard; and Radkowski, Peter P.: Numerical Analysis

of Unsymmetrical Bending of Shells of Revolution. ATAA Jd.,

vol. 1, no. 8, Aug. 1963, pp. 1833-1842.

Sepetoski, W. K.; Pearson, C. E.; Dingwell, I. W.; and Adkins,

A. W.:

Symmetric Thin-Shell Problem. Trans. ASME, Ser. E.: J. Appl.

Mech., vol. 29, no. 4, Dec. 1962, pp. 655-661.

Kalnins, A.: Analysis of Sheils of Revolution Subjected to

symmetrical and Nonsymmetrical Loads. Trans. ASME, Ser. E:

J. Appl. Mech., vol. 31, no. 3, Sept. 1964, pp. 467-476.

MN MN Radkowski, P. P.3; Davis, R. M.; and Bolduc, M. R.: Numerical

Analysis of Equations of Thin Shells of Revolution. ARS /d.,

vol. 32, no. 1, Jan. 1962, pp. 36-41.

ON Kalnins, A.: Free Vibration of Rotationally Symmetric Shells.

J. Acoust. Soc. Am., vol. 36, no. 7, July 1964, pp. 1355-1365.

Cohen, Gerald A.: Computer Analysis of Asymmetric Free

Vibrations of Ring-Stiffened Orthotropic Shells of Revolution.

ATAA J., vol. 3, no. 12, Dec. 1965, pp. 2305-2312.

Liepins, Atis A.: Free Vibrations of the Prestressed Toroidal

Membrane. ATAA J., vol. 3, no. 10, Oct. 1965, pp. 1924-1933.

LOL 102

Liepins, Atis A.: Flexural Vibrations of the Prestressed

Toroidal Shell. NASA CR-296, 1965.

Lo. Sanders, J. Lyell, Jr.: Nonlinear Theories for Thin Shells.

Quart. Appl. Math., vol. XXI, no. 1, April 1963, pp. 21-36. li. Stein, Manuel; and McElman, John A.: Buckling of Segments of

Toroidal Shells. ATAA J., vol. 3, no. 9, March 1965, pp.

1704-1709. le. McElman, John A.: Eccentrically Stiffened Shaliow Shelis of

Double Curvature. NASA TN D-3826, 1967.

13. Forsberg, Kevin: Influence of Boundary Conditions on the Modal

Characteristics of Thin Cylindrical Shells. ATAA J., vol. 2,

no. 12. 1964, pp. 2150-2157.

14. sobel, L. H.: Effects of Boundary Conditions on the Stability

of Cylinders Subject to Lateral and Axial Pressures. ATAA

Je, vol. 2, no. 8, 1964, pp. 1437-1440.

Hu, William C. L.; and Wah, Thein: Vibrations of Ring-Stiffened pu or

Cylindrical Shells - an "Exact" Method. SwRI Tech. Report No.

7, Oct. 1966.

16. Arnold, R. N.; and Warburton, G. B.: Flexural Vibrations of the

Walls of Thin Cylindrical Shells Having Freely Supported Ends.

Proc. Roy. Soc. (London), ser. A, vol. 197, no. 1049, June 7,

1949, pp. 238-256.

17. Arnold, R. N.; and Warburton, G. B.: The Flexural Vibrations of

Thin Cylinders. J. Proc. (A) Inst. Mech. Engs. (London), vol.

\ 167, no. 1, 1953, pp. 62-74. 103

18. Fung, Y. C.; Sechler, E. E.; and Kaplan, A.: On the Vibration of

Thin Cylindrical Shells Under Internal Pressure. J. Aeron.

Sci., Sept. 1957, pp. 650-660.

19. Koiter, W. T.: A Consistent First Approximation in the General

Theory of Thin Elastic Shells. Proceedings of the Symposium

on the Theory of Thin Elastic Shells, North-Holland Publishing

Co., Amsterdam, 1960.

20. Novozhilov, V. V.: Thin Shell Theory. P. Noordhoff LTD.,

Groningen - The Netherlands, 1964.

ol. Stein, Manuel: The Influence of Prebuckling Deformations and

Stresses on the Buckling of Perfect Cylinders. NASA TR R-190,

1964. XII. VITA

Mr. Cooper was born on June le, 1940 in Boston, Massachusetts.

He graduated from Boston Technical High School in 1957. He attended

Northeastern University receiving a B.S.M.E. degree in 1962. While

an undergraduate he was a member of Pi Tau Sigma, ASME and ASM. He

held a cooperative work assignment with Artisan Industries in Walthan,

Massachusetts from 1959 to 1962. He received a teaching assistantship

appointment in the Mechanical Engineering Department at Northeastern

University in 1962 and taught undergraduate courses until August 1964.

In June of 1964 he received an M.S.M.E. degree from Northeastern

University. In August of 1964 he joined the National Aeronautics and

space Administration at Langley Research Center where he is currently

employed. He is an associate member of ASME and is the author of two

technical papers, one in the field of metallurgy and one in the field

of shell dynamics and stability. He is married and has two children.

Gaul Ao Cuper—

104 XIII. APPENDIX A

COEFFICIENTS OF EQUATIONS (22), (23), (27), AND (90)

A. Coefficients of Equations (22)

The coefficients of terms in the governing equations (22) are defined as follows;

Fy, =2

Fig = 7 9

P (z-u) , ae (1 -y) 3k k ea, l 548 22 "3 96 ( @ 7 x) ye ( Eg)

2 XN (1 = p)n Fo3 = “Bir (3k - * )

Baa = Fy3

“30 = tos

PO NO 5 Ee u ry > I we oo one \N

105 106

(1 +u)n | An (1 - yp) (3%, - Kg) (3% - &) - nS (e + €,) Le er 96r

i i Kt uk, + X= w) (l+u)7 k,+ 13

(1 =u) 7% +e, 32 - at

Cay i M (2 =u")

el - Gis

1- dk x an 32 Bie 7K7 Kg (5 + Bu)- er (1 + u) ?. 2 | Ma)- a + u) (27K, ky + 77) + “y Dd 2 _ _ a * 7, + 20, =

LO7

- Sy

ne (a - uy

2 (1-4u)n e (1 = wp) 2,2 thy ~ HK, Kg - 7 - Dre 12 (1 +4) 7 K

n 2 (2, - Kg) 2 ob. pel ee * 2 Ke & 7 (e,, * eg] 8r Lr

dk x|— cor

+ la - 2) vk, + =| Pe ~ ky —(-uny 2 - ny | OX he er ler 3

+#(L4u) k kyl -%(e46,)- 88x 5, + Sewn | ee x *8 Ar ( x a)" pr Xx Pe Or @ Ve

oy + ¥(k k n° (1 = w)néy (3%, Ko) + (1 ) sale u dx ( x” ) ~ Lore 2 FH) kK

2 - KE (Ky + ul, | =Pe ,+ (1se -y)n =Pe hy x (a - a) (l-u )7k,, 108

log =

2 42 (1 - u) nm AX (1 -u)]{(L+u)n~, 2 22 "Gon + eo KL Rg - Te B Kg r r k_k =x 8 (3%, - “JF 2 +5 Yaffe(=, + _ea) -ky a €, - ZK,1 Kg

(1 5- wy (5, + 2k) ~Pe (i-y),2-2st ky” 9,

x (l= wy wey 62 2 8 ax

2 dk n N (1 - w)n Xx 2 2(1+ u)n 2) r (Ky + 4K) + oly Ya, 7 ey 2

(3k - k,) (7 2 + Kg) - ok,n _€y + (1 5-u)ny —

(1_- un, 5.24 (i= u)n Oe Br g Pe Dr ax

2 2 a w(a-u2)n Toy = 109

Hy = - 7 (Ky + HK) + no (17m =u) (lt+u)7| 7 kL - 7 o-- ak,one

2 ne dk, | + 2k Ma )* aa ry - 3 ay] - rk, +=] er

de (1 - p)n 2 2 , | dk x) 3,2 -R& +. Ore Sx | Peo mL + aR Pe

do - (2k O, + U7) oe E

Ho =~__ sRy (Kg t HK) + X (15 -p)n fata +) Pi 2 yk,

+ 27 2 Ky 2p)no te”di, 2 (Kg - k,) +k, ky O¥e - 5)

o.° un 9% - ~k,¢ *o € £9 , (h=u)n7By 5Pe _ 2 -u)n,Or ee "ry &

2 2 H.. =- kK 2 - Qik k, - ko”2,n + (1 -y)n (2 + unl, Ky = 8n + 2")2 D3 x x 6 9 or’ Q x

2 ne dk) _ +2(7 +k Kg] ~ Be - [7 (l + yu) + a= Pe

do - G-wn 3, - (Kk + uk) ae + 0 er é

H, -- (1 2) (1 )k_ k unt By ~ ~ He BY Ag T BT r

Hy - r= (2 - a) (= + 74, L1O

His = Hoy

nn (2 - 2) un’ i an,

irae D SN |

ae = 10n* @ - 2) f= ao

Coefficients of Equations (23)

The coefficients of the terms associated with the boundary conditions (eqs. (23)) are defined in the following equations:

ll 7

22 * D

"25 _é= ate A un (Fo x)

2 _ (1 =n _A - )n (3%, _ kg) (3%, - K,) + i (e,, + ee)

| M7 (Lo 8) (51, +E (E+ %) + Fe ae - Ue?

NY2 (1 - un (1 - p)n- - — tie —— (Pe ~ k,. | + —<3R .

2 (1 - up) 2 n 2 _ 12 fa ea yk tts Gx, - %) ~ kK &, er

-u7 B, =k, By

A - war P53, - k + 2(1+ 4) ky | - 3, 32 Me

A 2 (i - yu) ny2 _ 33 EEG 4) BE > (ty)

Poh Me (1 = yu) (2 - u)y

dle

fs == n @ - i) k,

C. Coefficients of Equation (27)

The coefficients of equation (27) are defined as follows:

Ky = oe : 5 4(855) 20(%15) - a55(®)o) + ecto A),

Fp Ay Boo 8s) F £8, AnzAoy, + (a5) Az ~ each as)

* (213)° = 4155] (24)

A= 822855 F 817859 7 817833 - (2) - Can - (3) | Buy

- (a2,)" (41, * 853) 7 (25) Crm Bop) + HB Pos)

A), = Hoy a55 7 Hes - Fog (3*) - " as, = Hay - Fy (EF) ayy = Hy

D. Coefficients of Equation (90)

The coefficients of equation (90) are defined as follows:

2 >i >i o = Puy Yosh - (%3)*| - Pip eaoPss - bashes

+ bis s2%es - b3Peo|

= Bopbss - (P55) + Pasha - (>43)° * DypPop - (40) url 114

REFINEMENT OF THE NUMERICAL PROCEDURE OF CHAPTER VIT

The method of Gaussian elimination used in Chapter VII introduces

spurious singularities into the determinantal equation (48). These

singularities are associated with sign changes of the determinant

even though no zero of the determinant exists for these values of 2.

This causes difficulties in the search procedure for finding the

frequencies. The actual value of the determinant of the coefficient

matrix of the set of equations (40), (41), and (42) may be written as PLP] A= AFo] >» Pane Fae Fe Fea (BL)

where the zeros of R are contained in the last determinant if

2a # O and in the next to last determinant if Zong = O (simply

supported). The last determinant is infinite at the frequencies of

a simply supported system since P is found by inverting a null N-1 matrix in equation (45). Similarly, the determinant containing the

zeros for the simply supported system tends towards infinity as the

frequencies for a simply supported system of length S-A is

approached.

To remove the singularities, equation (Bl) is used in place of equation (48). Although individual terms in (B1) will increase

115 116

without bound for certain values of the trial frequencies, R will remain bounded since there will always be corresponding terms approaching zero at the same rate. If Zu = QO, the last term in equation (Bl) is dropped and the modified R is used in place of equation (50). This procedure does not increase the time for calculation appreciably since the determinants required in equation

(Bl) have been found during the computational procedure followed in determining P, with equation (45). APPENDIX C

SOLUTION OF PRESTRESS EQUATION FOR CHAPTER VIII

The equations governing the axisymmetric prestress deformations

are found by applying equations (68) and (69) to equations (66) and

retaining the prestress terms aR, ) a = ° 2M d |= E98 Sra + at} == 0 >on

2 _— — d M, Ne No _ aw 2 7 R OR Me Tote 0 dé gE dé 7

The corresponding boundary conditions are ~

Ny -N or u-v

_ eM, _ _- - Neg + R = T or v=Ve=O from symmetry (C2) & — dw -— —- = ae + Ny aE = Q or weHW

—- = dw dW Me = M or ae ~= ae

dw where N, T, Q, and M are applied edge forces and U, V, W, and aE are applied displacement conditions.

The first of equations (C1) and (C2) and the second of equations (Cl) and (C2) yield

117

118

N, = N E (c3) _ 2M co = Neg + n= T

for all €

If no shear loading T is applied to the edges then

N . ee _ for all £9 R (C4)

The stress resultants may be written as

2, WW, = Ny =B(l- 4") g + HN

and the third equation in (C1) becomes with the introduction of the nondimensional variable x

ea 4b De 2 2 — XN dw Naw 2,-— _ pR (1 - ut) N Tee Bet hw) we eeGH) BG (C6)

Equation (C6) has the same form as the Foppl equation solved in reference 2l and in fact reduces to this equation when KL = 0. If the boundary conditions are the same on both edges of the shell, the solution to equation (C6) is

119

Ww . S 5 | Fea sina, (x -§) sima, (x 5

S S P - + Ao COScos & (x ~ S|2 cosh a 2 ( - 5)2) + “> _ 2 (C7) where

P= Rp (1 ~ un) ~-f{k tu N (c8) — Eh x B

and >

4 2 3N = = I - Lu +-_ — oy \2 -B (C9) _ ,]2 7 a5 = RK yp- ue - 2b NB J

For a simply supported boundary condition (W = M = 0)

BD 2 eo 5 S . Do, 2 A -( P \ a cos a, = cosh Bn 5 7 ea, an sin a, 5 sinh —-

1 Ll-u } 2 S 2 aSBp 2 § 2 aSBp 2 35 (sin = sinh = + cos a) 3 cosh “= |

(C10)

2 2 ~9 Bo 5 S ano2 A -P (a, -~ ay } sin a sinh 5 + 2a, a, cos 5 cosh a

° i - Le 2a a So sin’ “1°a § inh” f2P2 + cos se aBe cosh*SOSh #2°3 120

, - (2 a. 2 c 08 4 38 Sinh B2P ae sin “12& cosh a 228 ) ~ S . i - un a. 2 sin aeD cos a?a + Ay sinh Be?D cosh 2. 2

> (ca)

ao . fe) a. 2A?cos sinh 2 +a. 2 sin “2 8 cosh a 228 7 1 a S aS as a5 a. sin a cos = a sinh > cosh —5-

This solution could be used to determine the prestress quantities in the numerical solution of Chapter VII, in order that the effect of prestress deformations on the natural frequencies of the doubly curved shallow shells may be investigated. If there is no axially applied load (N = 0), then from equation (C6) the deformations and rotations are independent of Ke APPENDIX D

AXISYMMETRIC VIBRATIONS

For the particular case of axisymmetric vibrations (n = 0), the circumferential equilibrium equation, that is, the second of equations

(73), uncouples from the remaining equations hence the torsional frequency is independent of u and w. This equation, written in terms of the nondimensional meridional length x, becomes

2 2 1 -u N vhR _ ( 5 Gad)y, = Be Voz_ = 0 (D1)

Since KL does not occur in this equation, the torsional frequency will be independent of the meridional curvature.

The general solution of equation (Dl) is

. Qx Qx Lot vVo=j)v., sin "WC + v,. CURRcos e (D2) so that for the circumferential boundary conditions the torsional vibration frequencies is given by VAe4) (+) =ixn tle=tl1, 3; 5;

The minimum axisymmetric vibration frequency is thus given by

The remaining two equilibrium equations of equations (73) reduce to

uu, + (, + u) WwW, - yn” Uy, = 0

(kK, + u) us, + x Ws coe * (Ke + Qu k + 1) w- ne Woy (D4)

+ vn Wore = 0

These equations are handled in the same manner as were equations

(74). The characteristic roots of equations (D+) are determined from a sixth degree equation formed by equating the determinant of the matrix which results after deleting the second row and column of the

coefficient matrix of equation (76) to zero. With these modifications,

the terms B53 vanish as they should since there is no longer any inter- dependency between w and v. Since only six roots are present, the

sum of linear solutions in equations (82) range over six rather than

eight terms. With the dependence on v removed, the boundary

conditions in equations (85) associated with v and Neg must be deleted, that is, Noy Ney? Yay? and XG; leaving three boundary

conditions on each edge. The solution of the axisymmetric frequencies

is determined in the same manner as is indicated in Chapter VIII.

XVIT APPENDIX E

Printout of the three computer programs with sample output and flow diagrams used in the analysis.

1. Computer program for numerical method of solution of the deep

shell equations of Chapter VIT.

Computer program for general method of solution of the

approximate (shallow meridian) equations for Chapter VIII

for several boundary conditions.

Computer program for method of solution of the approximate

(shallow meridian) equations of Chapter VIII for a freely

supported shell.

123 12h

FLOW DIAGRAM OF MAIN PROGRAM FOR

NUMERICAL METHOD OF SOLUTION

OF THE DEEP SHELL EQUATIONS

START

Output

Constant Delta

Compute Variable Deltas

Output Variable Deltas

Begin loop on

mode numbers + DO 2500 IT = IX, IY

Are both Yes < IPLOT and >_> MODEPR = 0 No

126

; ?

CALL COOPER

Is Yes [ IJ = IFREQ(IK) 2500 Continue} —

No ' Read i Define AA( IK) new and BBB(IK) case to find next frequency

127

A. Main Program Variables

NCON - indicator for constant or variable delta values. IK - subscript associated with each frequency interval. IJ - counts the number of successive frequencies for a

Wh particular circumferential mode number. MAN - indicator for search method used.

* OMAGI ~ final interpolated value of the frequency. IPLOT - indicator for use of plotting routine. MODEPR - indicator for printing mode shapes.

COAATIAWF IFREQ - array containing number of frequencies desired for each circumferential mode number. AA, BBB - interval in which ITR2 subroutine searches for \o value of frequency.

B. Subroutines and Function Subprograms

1 NANCY - a user-written subroutine which supplies all geometry, prestress conditions, boundary conditions, and other input to the main program. 2. RAN - subroutine which calculates modified coefficient matrices of the differential equations and the difference equations. These matrices are independent of the frequency. ITRe - a NASA Langley Research Center library subroutine which, given F(X) = 0, searches for a sign change in a specified interval (A,B) and converges on X, using an interval halving procedure. ITR2 calls function subprogram FOFX, which in turn serves as a vehicle for calling COOPER. MANUAL - subroutine which calculates the residuals when given a frequency interval and a constant frequency increment. MANUAL calls FOFX, which calls COOPER. COOPER - subroutine which calculates the recursion matrices, the final characteristic determinant, and the mode shapes (if desired). Subroutines used by COOPER: a. MATINV - a NASA Langley Research Center library subroutine which finds the inverse of a matrix and calculates its determinant. b. ODDIPLT - a NASA Langley Research Center library subroutine which generates plots on tape to be processed off-line on film or oscillograph paper. 128 SNOTEVNOS

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*=39VINADUId *=39VINADUId

*SsJOVINSDUSd *SsJOVINSDUSd S=JOVINIDUSd S=JOVINIDUSd

*=JOVINADYSd *=JOVINADYSd (YADWNN (YADWNN HLIM VOIWE

(S)PDHOTSXS*B8°8T3I*=1S (S)PDHOTSXS*B8°8T3I*=1S

(HIDHOTSX4S (HIDHOTSX4S

(EMDHOTSX4*O°BSTA*=(E (EMDHOTSX4*O°BSTA*=(E

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BP BP B°8TI*=tZ B°8TI*=tZ

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TRUN=1 or 3

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——-W.

174

Define ot (3 and OME(IN) to find next frequency

26 Continue

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175

A. Main Program Variables

See comment cards in the main program for a description of of the input data. IN - subscript associated with each frequency interval. IFREQ - counts the number of successive frequencies for a particular circumferential mode number. N - circumferential mode number. QMAGI - interpolated value of the frequency. INT - counts the number of iterations in the ITR2 subroutine for each frequency.

B. Subroutines and Function Subprograms

1. MANUAL ~ subroutine which calculates the residuals when given a frequency interval and a constant frequency increment. Ce ITR2 - iterative halving subroutine which searches for a sign change and then proceeds to the frequency within a specified error limit. SHAPE - subroutine which calculates the mode shapes and stresses; results were not used in the thesis. NZERG - subroutine which calculates the minimum torsional frequency when the circumferential mode number is zero. RES - function subprogram for solving the characteristic equation, calculating the modal amplitude and boundary condition coefficients and evaluating the residuals; used for all circumferential mode numbers except zero. Subroutines used by RES: a. FALG - calculates the roots of the characteristic equation. b. ABCDI - examines and orders the roots of the characteristic equation. c. CDETERM - calculates the complex residual. ZN - function subprogram similar to RES; used only for circumferentiai mode number zero.

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METHOD OF SOLUTION OF THE

APPROXIMATE (SHALLOW MERIDIAN) EQUATIONS FOR A FREELY SUPPORTED SHELL

Read Input Data

K=(ALPHF-ALPHI) /DELTAtL t

Begin loop on

curvature parameter ft Dg 100 Iel,K

SAVE=10.E19

Begin loop on axial mode numbers "+ Dg 200 M=M1,MM

|! PMEGAM=10 E19

Begin loop on circumferential harmonic mode numbers DY 300 N=NL,NN

218 Y

Calculate coefficients of cubic equation

PMEGA=SQRT(ROPTS)

! y / 4 4 t Order @MEGAs from minimmto maximm

duican.cr. GMEGA(‘2) aie

No ANN=AN | PMEGAM=QMEGA (11)

300 CONTINUE

219 PG

Is SAVE.GT. @MEGAM ie

No SAVEN=ANN SAVEM=AM SAVE=(@MEGAM

200 CENTINUE

Put N,M,@MEGA into erreys A

Is ALPHA=0

No Yes

Is (ALPHA*LANR /2)**2.0T,1, —X&i_p¥

Calculate RISE DEEP and RISE SHALLQW

. *

100 CONTINUE |——_»»

220

2el

A. Main Program Variables

K - total number of curvature parameters considered. ALPHF - final curvature parameter. ALPHI - initial curvature parameter.

DELTA - difference between successive curvature parameters. FWY FWY OMEGA - frequency. M - axial mode number. N - circumferential harmonic mode number.

OANA OANA RGPTS - roots of cubic equation (frequency).

GMEGAM - minimum frequency within a specific M-loop. H Oro SAVE - minimum frequency.

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1. FALG - NASA Langley Research Center library subroutine which solves for the roots of an equation. 222 JTdWVS

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o0Sco°- o0Sco°- IV IV VIBRATION OF STRESSED SHELLS OF DOUBLE CURVATURE

Paul A. Cooper

ABSTRACT

Shells of double curvature are common structural elements in aerospace and related industries, but due to the complexity of their configurations and governing equations, little has been done to classify their general dynamic behavior. The subject of this dissertation is the determination of the effect of the meridional curvature on the natural vibrations of a class of axisymmetrically prestressed doubly curved shells of revolution.

A set of linear equations governing the infinitesimal vibrations of axisymmetrically prestressed shells is developed from Sander's nonlinear shell theory and both the in-plane inertia and prestress deformation effects are retained in the development. The equations derived are consistent with first-order thin-shell theory and can be used to describe the behavior of shells with arbitrary meridional configuration having moderately small prestress rotations.

A numerical procedure is given for solving the governing equations for the natural frequencies and associated mode shapes for a general shell of revolution with homogeneous boundary conditions.

The numerical procedure uses matrix methods in finite-difference form coupled with a Gaussian elimination to solve the governing eigenvalue problem. An approximate set of governing equations of motion with constant

coefficients which are based on shallowness of the meridian are

developed as an alternate more rapid method of solution and are solved

in an exact manner for all boundary conditions. The solutions of the

exact system of shell equations determined from the numerical procedure are used to determine the accuracy of the approximate

solutions and with its accuracy established, the approximate equations are used exclusively to generate results. The membrane and pure bending equations which correspond to the approximate set of equations are solved for a specific boundary condition.

The effect of the meridional curvature on the fundamental frequencies of a class of cylindrical-like shells with shallow meridional curvature and freely supported edges are investigated. Results

show that the positive Gaussian curvature shells have fundamental frequencies well above those of corresponding cylindrical shells. The fundamental frequencies of the negative Gaussian curvature shells generally are below those of the corresponding cylinders and evidence wide variations in value with large reductions in magnitude occuring at

certain critical curvatures. Comparison of the membrane, pure bending and complete shell analyses shows that these critical curvatures represent configurations at which the fundamental mode of vibration of the shell is in a state close to pure bending. The membrane theory affords a simple method of determining the modal wavelength ratio at which the pure bending state exists for a given negative Gaussian curvature shell, while the pure bending theory gives a good estimate

of the magnitude of the frequency for this wavelength ratio.

Meridional edge restraints and internal lateral pressure reduce the wide variation of the natural frequencies in the negative curvature

shells and in general raise the natural frequencies. External lateral pressure accentuates the reduction in natural frequencies of the negative curvature shells and causes instability at low compressive stress ratios.