Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00098431/00001
## Material Information- Title:
- Orthotropic cylindrical shells under dynamic loading
- Creator:
- Mangrum, Elmer, 1936-
- Place of Publication:
- Gainesville FL
- Publisher:
- [s.n.]
- Publication Date:
- 1969
- Copyright Date:
- 1969
- Language:
- English
- Physical Description:
- xvi, 152 leaves. : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Computer programs ( jstor )
Cylindrical shells ( jstor ) Damping ( jstor ) Eggshells ( jstor ) Equation roots ( jstor ) Inertia ( jstor ) Method of characteristics ( jstor ) Static loads ( jstor ) Structural deflection ( jstor ) Velocity ( jstor ) Buckling (Mechanics) ( lcsh ) Cylinders ( lcsh ) Strains and stresses ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis--University of Florida.
- Bibliography:
- Bibliography: leaves 149-152.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 025366980 ( AlephBibNum )
AFP3837 ( NOTIS ) 20143414 ( OCLC )
## UFDC Membership |

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ORTHOTROPIC CYLINDRICAL SHELLS UNDER DYNAMIC LOADING By ELMER MANGRUM, JR. A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THB DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1970 UNIVERSITY OF FLORIDA 3 122 08552 3057 3 1262 08552 3057 This dissertation is dedicated to my wife Rita and my daughter Gaila. ACKNOWLEDGMENT I would like to acknowledge the support and encouragement of General William M. Thames, K. A. Campbell, and N. E. Munch of the General Electric Company who made this research possible. I wish also to express my sincere gratitude to Dr. J. J. Burns for this guidance and sugges- tions during the course of this research. TABLE OF CONTENTS Page LIST OF TABLES .............. LIST OF FIGURES .............. KEY TO SYMBOLS ............. ABSTRACT . . . . . . . . . Chapter I INTRODUCTION .......... Statement of the Problem . . Specific Goals of This Research . Review of Previous Work . . Contributions of This Work . . II GOVERNING EQUATIONS OF MOTION General Equations ....... Axisymmetric Loading . . . Pressure Loading Form .... Nondimensional Equations . . III TRANSFORMATION OF EQUATIONS . IV INVESTIGATION OF THE CRITICAL VELOCITIES . . . . . . . vii . . . . . . viii . . . . . . xi xv ........... XV . . . . . .1 . . . . . . 2 . . . . . . 3 . . . . . . 3 . . . . . . 5 . . . . . . 5 . . . . . . 15 . . . . . . 15 . . . . . . 16 . . . . . . 20 20 . . . 23 TABLE OF CONTENTS (Continued) Chapter V SOLUTION FOR DISPLACEMENTS . . . . . . General Solution . . . . . . . . . Transformed Displacements . . . . . Inverse Transformation of the Rotation .. ... Inverse Transformation of the Radial Deflection Inverse Transformation of the Axial Deflection Summary of Deflection Expressions . . . Solution for No External Damping . . . . .. Form of the Radial Deflection in Region IV . . Form of the Radial Deflection in Region VII . . Form of the Radial Deflection in Region V . . Form of the Radial Deflection in Region VI . . Comparison of Solution with Other Results .. Numerical Results . . . . . . . . Comparison of Results with Other Solutions for a Static Load . . . . . . . . Summary of Deflection Response for Shells under Various Load Velocities . . . . . . Region II Response . . . . .. . . Region IV Response . . . . . . . Damping Effect on Regions V and VII Response Region VI Response . . . . . . . 2 Deflection Behavior in the Vicinity of X2CR 2 Effect of Prestress on 1 CR ......... Superposition of Step Loads . . . . . Study of Material Properties Variations . . v Page S. 42 42 S. 42 S. 43 46 51 S. 55 S. 58 S. 59 S. 62 S. 63 S. 63 . 64 S 65 S 65 S. 69 71 S 71 71 78 S. 78 78 . 82 82 TABLE OF CONTENTS (Continued) Chapter Page VI STRESSES . . . . . . . . ... .. ... . 92 Development of Stress Equations . . . . ... 92 Numerical Results . . . . . . . . . 94 VII CONCLUDING REMARKS ................. 98 Conclusions . . . . . . . . ... .. . 98 Suggestions for Future Work. . . . . . ... 99 Appendix A FOURIER TRANSFORM OF THE FORCING FUNCTION . . 100 B SOLUTION OF EQUATIONS FOR THE TRANSFORMED DEFLECTIONS . . . . . . . . . . . 102 C EVALUATION OF THE DISCRIMINANT OF A FOURTH ORDER POLYNOMIAL .. . .. .. .. .. . .. . . 106 D DETERMINATION OF THE CRITICAL VELOCITY EQUATIONS . . . . . . . . . . . 111 E COMPUTER PROGRAM FOR SOLUTION OF LOAD VELOCITIES WHICH CAUSE REPEATED ROOTS IN THE UNDAMPED CHARACTERISTIC EQUATION. . . . . . . . 116 F PARTIAL FRACTION EXPANSION OF A FOURTH ORDER POLYNOMIAL . .. . .. .. .. ... .. . 123 G CHECK TO SEE THAT SOLUTIONS SATISFY THE GOVERNING DIFFERENTIAL EQUATIONS . . . . . . ... 125 H COMPUTER PROGRAM FOR DEFLECTION AND STRESS CALCULATIONS ................... .. 130 I RELATIONSHIPS BETWEEN ANALYSIS PARAMETERS . . 147 BIBLIOGRAPHY . . . . . . . . . . . . . 149 ADDITIONAL REFERENCES ................... 151 LIST OF TABLES Page Correlation of Root Type with Region Numbers for Figures 4.2 through 4.8 . . . . . . . . . 27 H-1 Options Available for Program DEFSTR . . . . .. 131 Table LIST OF FIGURES Figure Page 2.1 Cylindrical Coordinate System. . . . . . . 5 2.2 Pressure Loading . . . . . . . . . 16 4.1 Flow Diagram of Computer Program VCRIT which Determines Load Velocities at which Repeated Roots Occur ...... 28 4.2 Classification of Roots of the Undamped Characteristic Equation for Variations in the Thickness-to-Radius Ratio Including Prestress . . . . . . . . . . 29 4.3 Classification of Roots of the Undamped Characteristic Equation for Variations in the Thickness-to-Radius Ratio with No Prestress . . . . . ... . . . ... . 30 4.4 Classification of Roots of the Undamped Characteristic Equation for Variations in E og/E o . . . . . . . . 31 4.5 Classification of Roots of the Undamped Characteristic Equation for Variations in Gxzo/Ex . . . . . . . .. 32 4.6 Classification of Roots of the Undamped Characteristic Equation for Variations in Evo/Exo . . . . . . . . 33 4.7 Classification of Roots of the Undamped Characteristic Equation for Variations in the Circumferential Prestress ...... 34 4.8 Classification of Roots of the Undamped Characteristic Equation for Variations in the Axial Prestress. . . . . . 35 4.9 Path of the Roots of the Undamped Characteristic Equation for Increasing Load Speed (Beginning in Region I) . . .. 37 4.10 Path of the Roots of the Undamped Characteristic Equation for Increasing Load Speed (Beginning in Region V) . . .. 39 5.1 Contour Integration Path for Evaluating Rotation Integral . 44 5.2 Integration Contour for Radial Deflection for ( > 0, bk > 0 47 5.3 Integration Contour for Evaluating Radial Deflection for p < 0, bk > 0 . . . . . . . . . 49 LIST OF FIGURES (Continued) Figure Page 5.4 Loci of the Roots of the Characteristic Equation as the Damping Approaches Zero . . . . . . .. 60 5.5 Static Load Problem . . . . . . . . . 64 5.6 Flow Diagram for Computer Program for Deflection and Stress Calculations . . . . . . . . . . 66 5.7 Radial Deflection for a Static Load on an Isotropic Shell (p = 0.3) . . . . . . . . . . . .. 67 5.8 Displacements for a Static Load on an Isotropic Shell (p = 0.3, h/R = 0.01) . . . . . . . . . .68 5.9 Radial Deflection Shape for Various Load Velocities . .. 70 5.10 Radial Deflection Response for Variations in Radial Damping (A2 = 2.0) . . . . . . . . . .72 5.11 Radial Deflection Pattern Immediately Above the First Critical Load Speed . . . . . . . . . . 73 5.12 Change in the Radial Deflection Pattern with Increasing Damping (?2 = 2.7) . . . . . . . . . .74 5.13 Deflection Wave Form at 2 = 5 and 10 . . . . .. 75 5.14 Effect of Damping for X2 = 500. . . . . . . .. 76 5.15 Effect of Damping for X2 = 2000 . . . . . ... 77 5.16 Deflection Response for Variations in Circumferential and Axial Prestress . . . .. . . .. .. .79 5.17 Maximum Radial Deflection in the Vicinity of the First Critical Load Speed . . . . . . . . . 80 5.18 Effect of the Axial Prestress on the First Critical Load Speed. 81 5.19 Variation of Pressure Pulse Length, d, at X = 30 . . .. 83 5.20 Response from a Smooth Sine Wave Type Pressure Pulse Using Superposition . . . . . . . . . . 89 5.21 Response from a Sharp Pressure Front Using Superposition 90 5.22 Radial Deflection Response for Variations in the Circum- ferential Modulus . . . . . . . . . . . 91 LIST OF FIGURES (Continued) Figure Bending Stress in an Isotropic Shell Under a Static Load (p = 0.3, d = 1) . . . . . . . . . . . Surface Stresses in an Isotropic Shell Under a Static Load (p = 0.3, h/R = 0.1) . . . . . . . . . . Page KEY TO SYMBOLS x, 0, z Ka ( = x, 0, z) R h U, V, W 0, 77 t No, Nap (ra,3 =x, z) Ma' Map (aCe =x, 8, z) Q( (a = x, 0) T N I h Pi PO C (a = x, 0) VY a (", = x, 0, z) CT ( = x, 0) Tr (&, p = x, 0, z) Exo E0o E10 coordinate axes unit vectors in coordinate directions radius of cylinder (to the middle surface) thickness of cylinder displacement in directions of coordinate axes rotations time stress resultants moment resultants shear force axial prestress stress resultant circumferential prestress stress resultant moment of inertia defined in Equation (2-7) initial lateral pressure mass density strain in a direction shear strain stress in a direction shear stress modulus in x direction modulus in 0 direction modulus in normal direction GOpo (a,p = x, 0, z) D (o = x, 0, i) Dx0 Ex (c = x, 0, v) Gx (c=x, 0, v) Gx0 I 2 Ka (a = x, 0) q'(x, 0, t) C.. 1j H(y) q V t U, W F P E 1 G E o r 0 I o E shear moduli defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) correction factors time varying lateral pressure load damping coefficient coefficients of a matrix Heaviside step function magnitude of lateral pressure load constant velocity constant defined in Equation (2-29) dimensionless displacements in axial and normal directions dimensionless axial distance dimensionless axial stress resultant dimensionless circumferential stress resultant dimensionless normal modulus dimensionless shear modulus dimensionless tangential modulus dimensionless inertia term dimensionless load velocity dimensionless rotatory inertia term dimensionless damping term r o D1 f(s) s c, c. (i = integer) C. D(s) e. (i = integer) f. (i = integer) 1 \i (i= 1, .... 8) Xi a b s. i, j AiCR VCR E qo c! (i = 5, 9) d Fk ak, bk R Res(a) a k' Ok thickness ratio depending upon pressure direction dimensionless constant transform of f(p), F[f(P)] = f(s) complex variable in transform space constants constants characteristic equation discriminant coefficients coefficients load speeds giving repeated roots in undamped characteristic equation approximate load speed roots constant, real part of complex root imaginary part of complex root nomenclature used to trace root loci the ith critical load speed dimensional critical load speed Young's modulus Poisson ratio dimensionless pressure load magnitude coefficients dimensionless load length functions defined in analysis real and imaginary part of complex root large radius defined in analysis residue of a function at point a coefficients in partial fraction expansions xiii , a K K, K' 1 2 1 /3 (0 )o (o = x, 0) (0 )i (ao= x, 0) Sk lengths defined in static problem analysis coefficients coefficient, defined in Equation (6-24) outer surface stress inner surface stress elements of determinant determinant of a matrix of coefficients integer Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ORTHOTROPIC CYLINDRICAL SHELLS UNDER DYNAMIC LOADING By Elmer Mangrum, Jr. August 1970 Chairman: Dr. J. J. Burns Major Department: Engineering Science and Mechanics An orthotropic right cylindrical shell is analyzed when subjected to a dis- continuous, finite length pressure load moving in the axial direction at constant velocity. The analysis utilizes linear, small deflection shell theory which in- cludes the effect of axial and circumferential prestress, transverse shear de- formation, and external radial damping. The problem is solved using Fourier transforms, and the inverse Fourier integrals are evaluated for the radial deflection, axial deflection and rotation by expanding the Characteristic Equation in partial fractions and using complex contour integration. By studying the discriminant of the undamped characteris- tic equation the load velocities which give repeated roots are determined. The loci of these load velocities separate regions in which the form of the displace- ment solutions differ. The behavior of these load velocity loci is studied for variations in the three nondimensionalized material moduli, the thickness-to- radius ratio, the axial prestress, and the circumferential prestress. By tracing the root loci of the undamped characteristic equation and by in- spection of the displacement expressions, it is determined that there are five critical load velocities (velocities at which the displacement becomes unbounded) for the specific example of an isotropic shell. An increase of the load velocity above the bar wave speed produces a deflection mode which is predominantly axial. The deflection response is investigated for numerous combinations of load speed, material properties, length of pressure load, axial and circumferential prestress, and radial damping. The axial prestress has a significant effect on the first critical velocity of the cylinder; initial compression tends to lower the velocity. Circumferential prestress has no pronounced effect on the critical load speeds but does influence the response at higher velocities. Variation of material properties was found to cause a rapid change in deflection response. Through superposition, the variation of pressure load length can be utilized to approximate the response to any desired pressure load. Examples of this application are demonstrated. A comparison of stresses and deflections against those predicted by the Timoshenko thin shell theory is shown for a static load. All of the above numerical work was done using dimensionless parameters which can be applied to thin shells in general. The calculations were done utiliz- ing a computer program developed from this research for the calculation of de- flections and stresses in the shells. The program is written in Fortran and is operable on the General Electric Company Mark II time sharing service. CHAPTER I INTRODUCTION Statement of the Problem One of the most commonly used geometries for structural application is the right circular cylindrical shell. This is particularly true in the aerospace field and in undersea exploration vehicles. In many aerospace applications the cylin- drical shell serves as the primary load carrying member for the rocket system and performs simultaneously as a portion of the pressurized fuel tank. In under- sea applications the quest for greater depth range has brought about many re- finements in structural optimization techniques. A result of the many stringent requirements being placed upon structural systems has resulted in two areas of rapid advancement: new material technologies and more sophisticated analysis techniques. The material technologies for advanced design applications have in many cases moved away from the isotropic materials and are utilizing orthotropic and anisotropic materials to satisfy the demanding requirements for more efficient, lighter weight vehicles. Studies such as that reported in Reference (1)* have shown that there is indeed an incentive for the application of these advanced technologies. Until recent years the mathematical complexity encountered when approach- ing the dynamic analysis of shells has been so formidable that few results were available for design applications. *Denotes entries in the Bibliography. The new technology demands mentioned previously have brought a response from the analysts in the past five to ten years and some of the more idealized dynamic shell problems have been investigated. The problem of particular in- terest in this work is that of a thin orthotropic cylindrical shell subjected to an axisymmetric pressure load moving in the axial direction. It is necessary to consider refinements to the theory such as the transverse shear deformation, axial inertia, and rotatory inertia effect so that the higher load velocities may be investigated. It is known that axial prestress has an influence on results in dynamic analyses. In this work the effect of axial as well as circumferential prestress is investigated. The specific loading considered will be a constant pressure pulse finite in both magnitude and distance which moves along the cylinder at velocity V. The shell theory utilized is linear, assuming small deflections, and by superposition it is possible to investigate the effect of various pressure pulse shapes. External radial damping is also included. Specific Goals of This Research The major goal of this research was to obtain a solution for the deflections and stresses associated with the problem outlined above. The secondary goal, although perhaps not secondary in importance to those interested in utilizing the results, was that of developing a computer program for the calculation of deflec- tions and stresses in the cylinder. Finally, the calculation and presentation of the effect of the many parameters included in the analysis conclude the goals to be reached in the study. Review of Previous Work A review of the early work on the response of a cylindrical shell to a moving load is given by Jones and Bhuta (2). Until the work by Nachbar (3), who con- sidered the dynamic response of an infinitely long cylindrical shell to a semi- infinite step pressure load, the axial inertia was not considered. Nachbar included the axial inertia effect and also assumed an external damping effect. However, due to the damping included, the first resonance condition was missed. Jones and Bhuta solved the problem with a ring load moving on an infinitely long cylinder but did not include the transverse shear effect. Other contributions were made by Reismann (4) who included the effect of axial prestress, which was significant as had been found in his work on plate strips (5). Hegemier (6) studied the stability problem for a large class of con- stant velocity moving loads but limited the velocity range to that lower than the first critical. All of the work previewed above was done for an isotropic material. More recently Herrmann and Baker (7) solved the problem of a moving ring load on a cylindrical sandwich shell of infinite length. Numerical results were presented for a core material which is assumed to have material damping. Also, the problem of a ring load moving on a viscoelastic cylinder was solved by Tang (8). Contributions of This Work The following contributions are believed to be original with this work. 1. Analysis of orthotropic monocoque cylindrical shells including trans- verse shear deformation, axial and rotatory inertia, radial damping, circumferential and axial prestress under a finite length step load. 2. Presentation of the forms of the solution with no damping for the seven- dimensional space whose coordinates are the thickness-to-radius ratio, 4 the three material property ratios, axial prestress, circumferential prestress, and the load velocity parameter. 3. Results indicating the effect of a finite length pressure pulse, and the capability to approximate any load shape through superposition. 4. Indication of the effect of prestress on the critical velocities of an orthotropic monocoque shell. 5. Results which show the effects of external damping throughout the load velocity range. CHAPTER I GOVERNING EQUATIONS OF MOTION General Equations A cylindrical shell of thickness h and mean radius R is referred to the co- ordinate system shown in Figure 2.1. Figure 2.1. Cylindrical Coordinate System Coordinate x is measured along the shell axis, 0 along the circumference and z is perpendicular to the middle surface. The unit vectors tangent to the coordinate lines at a point (x, 0, z) are designated by Kx, Kp, K The dis- placements in these three directions are ux, u and uz respectively. It is assumed that the displacements can be represented by the linear relationship in terms of z u (x, 0, z, t) = u(x, 0, t) + z4(x, 0, t) uo(X 0, , t) = v(x, 0, t) + Z7(x, 0, t) u (x, 0, z, t) = w(x, 0, t) + z z(x, 0, t) (2-1) where u, v, w are the displacements on the middle surface (z = 0); t denotes time; 0 and 7r are the rotations of a line perpendicular to the normal surface in the x-z and 0-z planes respectively. 0z is referred to as the thickness stretch. Equations (2-1) require that all straight lines normal to the middle surface of the shell before deformation remain straight after deformation. This is a good approximation if the shell is thin. Herrmann and Armenakas (9) derived a linearized theory for the motion of isotropic cylindrical shells subjected to a general state of initial stress by as- suming the final state of stress is reached by passing through an intermediate state, the state of initial stress. Subtracting the initial equilibrium equations from the non-linear equations of motion and then linearizing by disregarding all non-linear terms involving the additional stresses, the linearized equations of motion for a shell under initial stress are obtainable. Following this procedure, Baker and Herrmann (10) derived a linearized set of equations for the motion of orthotropic shells. Assuming an orthotropic cylindrical monocoque shell is under initial lateral pressure p., an axial tension T, and is subjected to a time dependent radial load, the five equations of motion have the form: Nx 1 O x u N 8u N 8w I + + T -+ hpu 0 (2-2) ax R ao aX2 2 2 Rax hp R ax R ao 1N aN Q xw aV 1 8 xe + 8v N 8w 8av " i + +-+T +2a = hpov R 0a ax R aX2 R 2 00a ax R\80 SaQ aw N 8v - -+-- -8 + h w+ N Q 2 + 2 R ax R aP 2 a2 0 a2 ) axx R 8 1 o)(w+ v) (h) h +2 + R R R a8 R + q'(x, 0, t) = (w + p hw ___ 1 __ Mx 1 Mx -X+- -x Q x ax R 8x x RM aM 1 80 x8M + -ax 0 R 80 8x '0 2 2 ) + au R2 a8 ax h o ( w R ax I . = I + u R 8w R) + (h -R)o7 30 o J I;R" R piR [1 + N + pR [i -] for external pressure (2-7) for internal pressure 3Po I h (2-8) These equations include the effect of external radial damping, axial and These equations include the effect of external radial damping, axial and rotatory inertia, and transverse shear deformation. (2-3) +h 1 o 8x (2-4) (2-5) Nh + 0 R2 R )h + o where (2-6) h = 0 IV- (1 8 The elongations, shears, and rotations have been assumed small in com- parison with unity. The strain-displacement relations are therefore taken in the form of Hooke's law. au Dx ax C i1 av , e R z o w) 1 8u + v Yx0 R + z aO ax Su aw xz 8z ax Sav 1 (w v) z az R + z \80 The stress strain relationships are assumed in the following form ax = Exo Cx + EV C x ox, VQ 6o (2-9) 0 = Evo x + E0o *xO = Gx0o YxO Oz = GOo YOz Txz = Gxo Yxz (2-10) Integrating the stresses through the shell thickness the z variable is eliminated and the stress and moment resultants are obtained. D = au + x x 8x R 8x = G + xo ax D a R 8x v Dx0 ex x0 ax R2 R D x au R 8x D 0 R 80 DX R D Gx R = G D + x0 8u R3 a- / D + D + - x 8x R w R 1 R +D " v ax 8e 8e + D x0 ax 1 R au 0 8x x6 ax G 0 - R + a R ag (2-11) D R2 +5 8 80 I Nx x0 0. + E au 80 v ax G R x S(G R +- v (Eo o R 8aw ax- ax + Do R3) (W where D = Ex I, D X x0 2' 0 = GxOo 2, Ex = E, h, o G = K2Gh G = K Goz h = E 0 12 D = El o 2' v o 2 = Ex h, E0 = E0o h Gx h, G = K Gxz h o x X o + h+ 2 2 12R2 S3 '2 12 The coefficients K and K0 are constants for adjustment and can be taken as x 0 7r/V2 as discussed by Mirsky and Herrmann (11). Substituting Equations (2-11) into the Equations of Motion, (2-2) through (2-6), yields 2 / E2 8 u x 8 v 8w 8 v E + + + x 2 R 2 R \ax ax8 8x ax 1 R [G a2v axa 0 Dx0 2 R 82O 2 80 (x+ + ) u1 R R ao ( GR 88 2 N u N 8w 2 2 R ax R a8 1 E R3D aw T(R R 3 a) 2 ax2 ax Gx R + 2 802 ) 2 8 u 8x80 IV - hpou + D 2o R3 82 R ae8 G 0 E 2 R ax8 R axa8 1 aw v + R 8 R I . = p hv + . 0 R (2-12) 2 + T 8_U 2 ax (2-13) D + R 2 axv ax a2v T 8 x 8x2 S8w 8a 2 + v 2 8a + )] (2-14) + h 1w 0 RiT a R2E + ) v+ R,) +'0 1 2w 18w N ho) ( R2 R + q'(x, 0, t) Dr a2 R 2 ax D R = w + p hw 2 +D C x 2x 8x 2 1 0x00 R2 (\2 R D 2 +D a2,1 R 8x80 Gx 8x - 1 a8v R 062] 2 Go D x 2 R 7 xOax +w ) ao V o aw + h + (h R)r = R 10 0 (ho Collecting coefficients on deflections gives R + [ (E + S[Dx 2 R ax 2 ax 2 8G 1 + G ) v + (E N) w x0 8x8 I R V ax DxO a2 R3 002 B 80 21 I 8t I a2 = 0 T at2 + -Go R 80 E Sa8u R ax + D R3 ao 8v ao) + + ao ) h (1 2 8w + T ax2 hR R 8 80 Q + G I- ax +Rh o ax +R u ax (2-151 802 1 R Nh o R I " SI +-R u (2-16) D aV R 8x80 + xe0 /ax R \axa o ax2 ) Nh [ SR2 " I 77+ - R E x2 E x2 (2-17) 82 + T a2 2 ax N 2 + o h 2 2 0o R a0 32]u 2 at (2-18) + 2 ax2) 8 '0 8 0 N av R2 o + aw ax + x/ 802 R ) 0 a + + N + (Gx + T) 2 2 xx 2 R 8 a ax G 2 ] - -p h- v R2 o at2 + 1+ w Go + 2 R D 2 G hN x8 a8 o + ax2+ R+ R 8x R I 2 I a2 Rat2 (E +- D 0 R Go R2 (E + 0 ( a+ at Do h \ R2 R+ p0a 2 atI + + RG0 D a2 Dx a2 R ax2 R3 a02 G) + D + Gx+ D a / -ax2 2 a 1 (G + T) 2 (G x 2 20 ax R SNh a ] x R ax = q'(x, 0, t) I a2 R t2 at Dx0 2 aa 2 2 2- R ao at + [ .(D +DxO) a ] = 0 a[ 0 + D 2 Ra 8 3 2 R ao 1 L (Ev + [ 1 R [ R 1 [R2 (2-19) h + R 2 ao + N) 80 (2-20) + Nh R ( -N) -L ax (2-21) 2 1 8 1 (E + G ) u + ( E Ril v x0 axaoI 2 0 1 (E R 2 0 a0 R2 T Rl + Nho 1 + S+ Nh ) G x R ax 13 [D 2 D a2 Nh \ a2 x0 8a 0 8 1 o I R R ax2 R3 a2 R R at + [Do Nh ( + + -+ G + +- w 1 2 + [ (D + DxO) D 2 S 2 D 2 Nh a ax R 2 0 at S + Go ( -R + 1 -I = 0 (2-22) Equations (2-16), (2-17), and (2-18) express the principle of linear momen- tum in the x, 0, and z directions, respectively, while Equations (2-21) and (2-22) express the principle of angular momentum about an axis through the middle surface in the direction of K0 and Kx, respectively. These equations can be written in the form C.. u. = q, C.. = C.. (2-23) 1ij J i 1J Ji In matrix notation the set has the form C11 C C13 C14 0 u 0 C C C 0 C v 0 12 22 23 0 C25 v C13 C3 C33 C34 C35 w = q'(x, 0, t) (2-24) C14 0 C34 C44 C5 4 5 0 C25 C35 C45 C55 where the coefficients are defined as follows: a2 = (E + T) ax + (G R Sa2 a2 + N a -Poh / ao at 12 C 13 D 2 C x a 14 R x 2 ax D 2 R3 ao2 R ao = (Gx +T) xO 2 2 ax D R2 R 2 1 E 23 R2 D 2 R Sh 0 +G +N 1+_ 0 R a2 Do 22 a2 2 ax R a0 = -(G + T) x 2 a ax2 G Nh 0 o +-+ R2 1 R2 (G0 R2 0 2 +N) a 802 I a2 R 2 R at2 1 (E R D h +-Do + N- R2 1R R a a2 + t p h 2 aat2 ( h\ G + N h 1 6 2 D0 HR RLe a2 C = D 44 X x 2 1 C45 R (D S a2 = D ax +D ) a x0 axa8 1 2 I (E + Gx ax R v x0 axao 1 a R (Ev -ax Dx 2 R 2 I a R at2 C22 2 2 ao G a2 R -oth R 0 at2 Dx0 R C 25 C 33 C 34 C a x hR h -N- R + Nh (1 0 -G x a2 a02 a J Q a2 at- at' C 55 + D82 R2 a2 R ao h ( -2 1 R -G, -N 0 h) R -4 at at2 (2-25) + -R E Dx0 + R2 Axisymmetric Loading Assuming the loading on the cylinder is axisymmetric, the set of Equations (2-24) reduces to the following. (E + T) p h XR2 2 0 1a I 2u -(E -N) -oh ax x + (E + D- + N Dx a2 u I 82 u R ax2 R at2 Gx+ + N = S hR S+ (E N) -- + I = 0 at2 R ( ax R ax2 R at2 2 2 w aw 8 w +T) a2 + ph ax2 at o at2 w- + N ) = q'(x, t) R x R x + N o aw +D a2 i 2 Sx R x ax 2 at2 / Ox at 0 Pressure Loading Form A step input in external pressure which is finite in both magnitude and time and which travels down the length of the cylinder at constant velocity V can be represented in the form q'(x, t) = q{H[Vt x] H[V(t t ) x]} (2-27) where H(y) is the Heaviside step function, defined as 0, y < 0 H(y) = (2-28) 1, y>0 (2-26) 16 This pressure loading is represented schematically in Figure 2.2. q --- V R 0 0 x Figure 2.2. Pressure Loading Nondimensional Equations The steady state solution will be investigated. Making the transformation a = x Vt (2-29) the partial derivatives may be written in terms of a. ax a8x aa a a a =-V- at aa at aa S= 2 (2-30) at a82 Using these relationships in Equations (2-26), letting u U-U R w W -( R a R (2-31) multiplying by R/Ex in the first two equations and 1/Ex in the third gives T dU + d2U x d/ 2 IV2 E R2 x Oh V E x dU d02 E + E S D + x 2 RE x 2 dp_ d62 d 2 = 0 do2 E V N dU EEx dx d + + -- E R2 x + E x / d2W d 02 _VR dW E d x oh V2 E x d W dp2 + h ) N ER x [H(-R)) H(-Rp Vt )] 0 V2I d2U R2 E d 2 x N h\o dW E R d IV2 d2 E R2 dq2 x Gx E x N ho =0 + Ex ER x /G x E Sx N h +E - E R x = q E x dU d (2 D x R2 E x D + x R2 E x d22 d02 (2-32) N ) dW SE x d Now the T E x N x E E x E E x h 0 R 18 following dimensionless ratios are defined. = F -P = o G E D -x E 0 - E G, E - 1' Ex o Ex ER Ex x R (1 + = r o vYp R 2 O R E x Po h P R o I I R p R3 o h3 12R3 P 12R3 p o, h3 12R3 2 = (2-33) E p x o The dimensionless velocity parameter is denoted by X and r refers to the inertia. A distinction will be made in the axial and radial inertia terms for later investigation, r denoting axial and r denoting the radial inertia terms. D x = D R2 E x Using the dimensionless ratios given in Equations (2-33), the equations of motion now take the form (1 + F r ) + (E P) + (D I 2) 1 d$ (P2 1 d 1 0o dU (E ) - 1 d 2 d2 d (P 2 = 0 (2-34a) (G+r2) d2 V dWV (G+ F -rA ) d X + (E +Pr )W d2 d4 o o0 - (G + P ro d -qo[H(-R) H(-R Vto)] (D -I x2) d (G+P 1 0 dP2 .ddw - (G+P r0) = 0 (2-34b) (2-34c) CHAPTER III TRANSFORMATION OF EQUATIONS Equations (2-34) will be transformed using the Fourier transform oo F[f(cP)] = f(s) =f f() e-is" dcp (3-1) The inverse transform is 00oo F [f(s)] = f(f) = f f(s) e's ds (3-2) where i = v-T. Assuming all the derivatives of f(o) through order (r-1) vanish as - +o the transforms of derivatives of f(o) are given by -k k - f (s) = (is) f(s) (3-3) so SO F d is f(s) (3-4) F -= -s f(s) (3-5) Applying this to Equations (2-34) gives 2 2- 2 2- (1 + F r I2) s2 (s) i(E P)s W (s) + (D I \2) s2 v(s) i(E P) s U(s) + (G + F r X2) s W(s) iE s (s) o q isd + (E + P r) W (s) i(G + P r) s (s) [ e 1] (D I X2) 2 U(s) + i(G + P r ) sW (s) + (D I X2) s2 (s) 1 0 0 1 0 + (G + Pr ) I(s) = 0 = 0 (3-6) Collecting coefficients on like displacements allows this set of equations to be written in the following matrix notation. [(1+F-r l2 )s2] [-i(E -P)s] 1 1 [i(E1 -P)s] [(D1 -I X)s 2] [(G+F-r X)s -i EXs +(E +Pr )] [i(G+Pr )s] o [(D Io )s2] 1 0 [-i(G+Pro)s] [(D1 I 2)s2 +(G+Pr r)] 0 *See Appendix A for the derivation of the transform of the forcing function. U(s) W(s) (s) 0 o isd - - 0 (3-7) 22 This set of equations can now be solved for U, W, and b. This work is carried out in Appendix B and the results are isd (s) = (1- e [(D I 2) (G+Pr +E P)s+ (G+Pr )(E -P)] q 2- 1 0 0 1 0 1 o s D(s) S1 -isd W )= (1--e [(D I 2)cs2 +(G+Pr )( + F-r l2)] o isD(s) isd = (- ) [(E -P)(D -I 2) + (G+Pr )(l+F-r A)] q 1 1 0 0 1 (3-8) where S4 3 2 D(s) = c +i c s +c s +i c s +c (3-9) 4 3 2 1 O and c = (G+Pro) [(1 + F-r1 X)(E +Pr) (E1 P)] c = E X(G+P ro)(1 + F- r 12) 1 0 1 c = (D I1 X2) {(E0 + r)c (E1 P)[(E1 P) + 2(G + P r)1 } +(G+Pro)(1+F- r 2)(F-Pr r 2) c = E X(D1 I X2)c 3 10 c4 = (G+F-r A)(D 1 X)c 2 2 c = 1+F-D -rA +I_ A S 1 + 1 r1 X +1o X (3-10) The displacements are found by inverting Equations (3-8) using transforma- tion (3-2). In order to evaluate these integrals the roots of the characteristic equation D(s) = 0 must be determined. CHAPTER IV INVESTIGATION OF THE CRITICAL VELOCITIES It is known that the inverse of the deflections [given by integral (3-2), where f(s) represents the deflection expressions (3-8)] does not exist when there are repeated roots of the characteristic equation D(s) = 0 on the real axis. This can occur when there is no external damping and corres- ponds to a resonant condition as discussed by Jones and Bhuta (2). There are specific load velocities corresponding to these points and they will be referred to as critical velocities. The condition which must be satisfied in order to have re- peated roots is that the discriminant of the undamped characteristic equation c s + c 2 + c = 0 (4-1) 4 2 0 must be zero. The discriminant of this equation is determined in Appendix C as 2 2 A = 16 c 4(c2 4c ) (4-2) 04 2 04 Therefore, the three conditions which will make the discriminant zero are c 4c c = 0 (4-3) 2 0 4 c = 0 (4-4) c = 0 (4-5) Substituting the required coefficients from (3-10) into (4-3) gives* C + CA6 + C4 + C2 + C = 0 (4-6) *See Appendix D for the detailed calculations. *See Appendix D for the detailed calculations. where The 2 C = e -e 8 9 14 C = 2e e e 6 89 C = e2 + 2e e e 4 8 7 9 12 C = 2e e e 2 C = e e 0 7 10 e. are defined as e = D e + e f f 7 1 5 302 e = D e e I e ef 8 1 0 4 0 5 35 e = e rr e e 9 3 1 O 0 4 e = 4e e f D f 10 3 63 1 1 e11 = 4e3[e6(f3f9 -rDf1) eo rlafDf e 2= -4e[e (r f +f I e) + eo r (f f rD f)] e3 = 4e [e6rI oe + e r (rf + fIe )] e4 = -4e eO r rl e4 14 3 0 1 0 4 (4-7) (4-8) and the f. coefficients are defined in Appendix D. 1 If the axial and rotatory inertia are neglected the equation for determining the location of the repeated roots reduces to X + C X2 + C = 0 (4-9) 4 2 0 where C = e2 r2 2 4 3 0 C = 2re 2eD f f (D e +e f f)] 2 3 6 11 o 1 5 30 2 X 2 C = (D e +e ff ) 4e f D f e 0O 15 3 02 33 11 6 (4-10) The solution of Equation (4-9) gives two roots (X' )2 and (X' )2 which are 1 2 velocities at which repeated roots occur. The second condition which gives repeated roots is that given by Equatic (4-4). Substituting for the c4 coefficient gives (G+ F r?2) (D I A2) [1 +F- D + (I r) ] = 0 (4 1 0 1 0 1 -11) This can be written (f3 r2) (D1 I X2) (f + e4 2) = 0 Expanding this and collecting coefficients on 2 yields [f D -(rD +f I) 2 + I r4] (f + e4 2) 3 1 1 30 0 1 f f3 D + [ e f3D f (rD1 + f3 2 + [f rI e (rD +fI )]4 + eI rX6 1 0 4 1 30 40 (4-12) = 0 = 0 Finally C' X + C 4 + C X2 + C = 0 6 4 2 0 where C = e I r 6 40 C = f rI e (rD +f I 4 10 4 1 3 C' = e f D f (r D + f3 I1 2 4 3 (4-13) C' = f f D 0 13 1 )n (4-14) If rotatory and axial inertia are neglected Equation (4-13) reduces to give one root C' f (X)2= -- = r (G + F) (4-15) CtC r r 2 The third condition (c = 0) from Equation (4-5) is now investigated. Substituting for co gives o2 (1+F- r 2) (E+ Pro) (E- P)2 = 0 (4-16) or 2 2 (f r ) e = e (4-17) 0 1 O 1 thus e f e 2 00 1 "4 e r o 1 X4 = er (4-18) 4 e r o 1 Therefore, Equations (4-6), (4-13) and (4-18) can be solved to give eight roots in X2 which satisfy the conditions for repeated roots. Equation (4-6) gives four values of X2 and these values are labeled X, X2, X2, and X. The condi- tion which lead to these roots was that c 4c c = 0 2 0 4 Solving Equation (4-1) directly gives c c2 4c S= 2 2 4 (4-19) 2c4 If the radical in (4-19) is zero, the roots are c c 2 2 s c2 s (4-20) 1,2 2c 3,4 e4 4 4 The repeated roots are either real or imaginary depending upon the sign of the coefficients. 2 2 Equation (4-13) gives three more roots which will be labeled X XA, and X These roots come from the statement that c = 0 and if c2 0 the repeated roots in this case will occur at an infinite value. Equation (4-18) adds one more root, making a total of eight. This root is 2 designated A3. The condition leading to this root was that c = 0. From Equa- tion (4-1) it is observable that the characteristic equation becomes s2 (4 2 + c ) = 0 (4-21) which shows a repeated root at the origin. A computer program was written for the solution of these equations. A simplified flow diagram of the program is shown in Figure 4. 1 and the details of the program, named VCRIT, are presented in Appendix E. The results of a parametric study using the computer program VCRIT are presented in Figures 4. 2 through 4. 8. The locus of each of the roots Ax (i = 1, .. .8) is shown on these figures. These curves are the boundaries which separate these plots into distinct regions which are labeled as Regions I through VIII. In each of the regions the roots of the undamped characteristic equation have a par- ticular form as noted on the figures and as listed in Table 4-1. Table 4-1 Correlation of Root Type with Region Numbers for Figures 4.2 through 4.8 Region Form of Root I, III, VI ibl, ib2 I a + ib, -a + ib IV, VII + a, t a2 V, VIII a, ib a, b are real Figure 4. 1. Flow Diagram of Computer Program VCRIT which Determines Load Velocities at which Repeated Roots Occur Insert Data in Separate File h EOo Gxzo E0o P, F R' Exo Exo 'Exo Pure 10.000 1.000 100 10 2 1 0.1 0.01 0.001 -- 0.00001 0.0001 0.001 = 0 Gxzo 0.35 Eo F = 0.002 2 Real, 2 Imaginary Roots 0.01 Figure 4.2. Classification of Roots of the Undamped Characteristic Equation for Variations in the Thickness to Radius Ratio Including Prestress 10,000 1000 t=0 Gxzo - = 0.35 Exo Pure IV Imaginary 10 Roots 2 Real, 2 2 Imaginary Roots X Loci, i= 2 0..00001 .0001 .001 ..01 0.1 h/R Figure 4. 3. Classification of Roots of the Undamped Characteristic Equation for Variations in the Thickness to Radius Ratio with No Prestress .. ...... ....... ...... ........ ............ r e s t r e s Pure Imaginary Roots 2 Real, 2 Imaginary Roots 0.40 0.80 1.2 1.6 E0 /Exo Figure 4.4. Classification of Roots of the Undamped Characteristic Equation for Variations in Eog/Exo 10,000 1000 100 A2 Roots 10 I 0.1 0.01 .001 .00001 Figure 4.5. 0001 001 .01 0.1 Gxz/Ex xzo x~o Classification of Roots of the Undamped Characteristic Equation for Variations in Gxzo/Exo VII Real Roots 1000 100 10 1 0.001 0.01 Evo/Exo Figure 4.6. Classification of Roots of the Undamped Characteristic Equation for Variations in the Evo/Ex Complex V lotsoo VI P, Pure : . .. . . : *2 R . S... . .: Imaginary Roots Real Roots IV 2 Complex Roots U \i Loci, i = Re 1. .'e 2 n"imaginary SRegion Root s ..-- .,,,.. ., .. hR- = -. (01 :- " - .mag inar' F n.nnl v' X R oots -E Po E = 0 n ',,: -' n ". ,. ' I.,,,. : ' , :v , .,. 0.10 0.01 0.001 0 0001' 0.0001 1000 100 Region I i 2 .....Imnaginary P .. Roots...... |: : a::: { :: :: :::::::::: ::::: == ==========:::, :.,1 :: .......:::::::::::::: .. I ?, k.:: :: ::: :i':<.:: ::: 4 :~i::. ..: .:: :::: ::::::::::::::::: :::::. :::::: :::::: :::::: ::::::::! f: : ::-: : ::: :::::::::::::::::::::::::::: ::: : :: ::::: :::::::::::::: ::::::::: ::::: : :::::::::: -;} |:::! ..:: .: ..: ....:.... :: i `.. ':* <.:!| ...... ....:" ;. 0.002 0.004 Figure 4.7. Classification of Roots of the Undamped Characteristic Equation for Variations in the Circumferential Prestress VII (Real Roots) .11. ... .. .. ...... .. . .. .. |. ..|.. .. v . 9 [LYmagiinar - 0 E O M -N ReI R t c1 o=t S L 0 F 0 F C Loci, i =1 :::ffil m :~::::-, .0 .35 1.3 1.001 1.002 3 21 4ii~ -0.002 Real Roots VII 1000 100 10 5 A' 2 1.0 0.1 -0.002 -0.001 0.001 0.002 0.003 0.004 F Figure 4.8. Classification of Roots of the Undamped Characteristic Equation for Variations in the Axial Prestress 2 Real, 2 Raginary Imaginary Roots Roots > h = 0.001 - E 0 = 1.0 E Xo = 00.35 P =0 Real Roots = 0 100 Complex .'"... Soo .o Roots ; %%, , "'. ". 7 ... ": ,. . ,: . ".;. '. .. :. ., .. ,, ,".] :: .? ;.,, # ix ,, % ..%~ 2 :' ..'.', ,,', d', ''"''' _."_'; ':: ... :-. .. ..' .,' ... .. "'". l 0.01 The lower three values of X. are approximated by Expressions (4-9) and 1 (4-15) by neglecting the axial and rotatory inertia. The approximations are quite good over the range of parameters studied. The critical load speeds are denoted as those speeds at which the displace- ments become unbounded for an undamped system. This corresponds to the load speeds which produce a double root on the real axis as will be shown later. It is instructive to follow the path of the roots of the undamped characteristic equation in the complex plane as the load speed increases. As an example, the roots will be traced for material properties corresponding to an isotropic shell with h/R = 0. 001 and positive axial and circumferential prestress. Following the vertical line for h/R = 0.001 in Figure 4.2 for increasing X2 will give a path crossing all of the boundaries separating different types of roots. Starting at the low load speed, the roots are all on the imaginary axis. (This would not be the case if prestress were not included, as shown by Fig- ure 4.3.) As a means of tracing the location of the roots, Figure 4.9 is util- ized which shows the complex plane. The roots in Region I appear on the plane on the imaginary axis and these particular roots are designated s ,1 s2,1' s ,, and S ,. The nomenclature s. denotes the ith root location and j indi- 3,l 4,l 1,j cates the relative position of the roots. For instance s,4 denotes the position of the first root and at that time (load speed) the other roots are located at points s 2,4 s and s The arrows indicate the direction in which the root is moving for an increasing load speed. At the first speed two pairs of roots are moving toward one another on the positive and negative parts of the imaginary axis. They meet, and the first repeated root location is established which cor- responds to ,1 in Figure 4.2. As the load speed increases Region II is entered. The roots are complex as can be seen in Figure 4.9. Next, the complex roots approach one another in pairs on the negative and positive real axis. This gives 37 the first repeated roots on the real axis and this load speed is designated as the first critical load speed, hXCR. / / / 3,2 [ .3, v1,3 43,r 3,- 3, S=A CR A1CR 1,5 Complex s Plane 4,i1 I Re(s) I / / / / / ' 54,4 8i'j = Root I at load speed j Figure 4. 9. Path of the Roots of the Undamped Characteristic Equation for Increasing Load Speed (Beginning in Region I) // Now the roots separate and go in opposite directions along the real axis, the larger roots eventually becoming unbounded. This speed corresponds to the boundary line between Regions IV and V. The condition causing this occurrence is that c4 -) 0. Repeated roots occur at infinity and the second critical load speed has been determined. As the load speed increases further, the large roots come in from ico along the imaginary axis. At location 5 the two large roots are imaginary and the other two are still on the real axis approaching the origin. This corresponds to Region V in Figure 4.2. The locations of the roots are.now transferred to Figure 4.10 to avoid undue complication of the picture. Two roots meet at the origin while the other two are yet large and imaginary. This double root corresponds to A4 in Figure 4.2 and is the third critical load speed. An increase in load speed now causes the roots to become all imaginary which is in Region VI. The last four values of 2 X. (i = 5, 6, 7, 8) are very close together. In fact 2 is approximately 1000, and 1 5 2 2 2 X6, A7, and A are almost nondistinguishable at 1002. Increasing the load speed 2 past A5 causes the large imaginary roots to proceed to an unbounded imaginary value and reappear on the real axis so that the roots are now two real and two imaginary. Also As must be a critical speed because Condition (4-4) is satisfied at load speed As. This region is not distinguishable on Figures 4. 2 through 4.8 and is not given a number. Further increase in load speed moves the imaginary roots out and eventually they reappear on the real axis with the other pair, there to remain. This posi- tion is indicated by s (i = 1, 2, 3, 4). It is interesting to note that A7,8 1,10 6,7,8 also must correspond to a critical load speed, XsCR. The lowest critical load speed for an isotropic shell is therefore XCR = X2. Figure 4.5 shows that for a material with a very low shear modulus, the first critical speed becomes 3 = A2CR which corresponds to the shear wave speed. Im ( s) I cu s 40 1.5 A. C4C11 S 3.- S S S S S S 4,. 4_ 1031 2 '.10 1,. / / / I I -LT-pt~-S S J s < 4.- -i A . 4.A 4, -I Complex s Plane \ S S SHeks) I / / / CR -\ Root i at load speed j Figure 4.10. Path of the Roots of the Undamped Characteristic Equation for Increasing Load Speed (Beginning in Region V) /. A S s 2. S 2. s 2. S 3.1 ) Y_ Y_ __ _____ _Y _Y A, -. s 4 40 When this load speed goes to zero, the critical small deflection buckling load has been reached. From Equation (4-15) this load is seen to be approximately F = -G (4-22) This corresponds to the second critical load for engineering materials with a more realistic value for shear modulus. If prestress is not included, Region I disappears as shown by comparing Figures 4.2 and 4.3. Decreasing the tangential modulus drastically or increas- ing the normal modulus results in the same effect as shown by Figures 4.4 and 4. 6. Circumferential prestress has essentially no effect on the critical load speeds but axial prestress has a pronounced effect as shown by Figures 4. 7 and 4. 8. It is instructive to make a comparison of these results with those obtained in Reference (2). Three critical velocities were derived there, which are given as -2 Eh Eh 22 VCR2 2 2 p R[3(1- )]2 6pR (1- ) 2 E VCR, = Po(l - ) 2 E VCR3 - 0ol_# (4-23) (4-24) (4-25) for an isotropic material. Since, in the nomenclature used in the present work, V oR(1 p2) = Eh (4-26) The corresponding expressions in terms of 2 are /2 P h XCR = 6 R 2 R ACR2 h R2 R (I- 2) XCR3 hW(1- ) (4-27) (4-28) (4-29) Now taking p = 3 it is found that 2 h XCR = 0.55075 0.15 R (4-30) hCR h (4-31) 2 R XCR3 = 0.91- h2 2 h From Figure 4. 3 (which is a plot of Xi versus R for an isotropic material with p = 3) for the case of zero axial and circumferential initial stress, XCR1, 2 2 2 2 XCR2, and 2CR3 from Reference 2 agree extremely well with 2A, X and 3 2 5B, 6,7, ' XA respectively. The two velocities, X and X, arise in the present results be- cause of initial prestress considerations and shear deformation, respectively, which were not included in the referenced results. X2 corresponds to the dili- tational wave speed, X corresponds to the shear wave speed, X4 corresponds to the bar wave speed, and 6,7, corresponds to the plate wave speed. 5,b,7 ,8 CHAPTER V SOLUTION FOR DISPLACEMENTS General Solution Transformed Displacements From Equations (3-8) the transformed displacements can be written as isd c' s + c' U(s) 1- e 5 6 Ss2 D(s) isd c s2 + c' W(s) 1 -e 7 8 q is D(s) cI =- (1 eis) (5-1) qo D(s) where c' = f (e + e) 5 73 1 cl = e e 6 3 1 c = f(f +e X2) 7 7f1 4 c' = e (f r x2) 8 3 0 1 c' = e f +e (f -r X2) 9 1 7 3 0 1 f = D -I 7 1 o (5-2) Inverse Transformation of the Rotation Using the inverse transformation given by Equation (3-2) the rotation is defined as c' 9 27r isd J (1 eisd) ei(s ds -0e d(s) __ Dfs) By partial fraction expansion this can be put in the form* (5-3) (1 eisd) eis 4 s ds k= 1 1 S[Ds(s)] d s S : k 1 4 m k m= m=1 9 c = 9 c 4 Defining c k 27r 00 isd (1-e ) i( ds s sk (5-5) (5-6) S= Fk(E ) k=l *See Appendix F for the derivation of a sample partial fraction expansion expression. (5-7) c 9 ~ 27r -CO where (5-4) then qo (sk sm) Letting isd 1-e sd f (s) _-e eis (5-8) s s k the integral to be evaluated is 00 f fk(s) ds o00 where fk(s) is an analytic function except at the simple pole s = sk. Defining the complex root in general to be sk = ak + i bk ' the Cauchy integral theorem is used to evaluate this integral. Im(s) C-F -a Re(s) Figure 5.1. Contour Integration Path for Evaluating Rotation Integral Assuming first that bk > 0 the Cauchy integral theorem gives Sfk(s) ds = 27T i Z Residues (5-9) (5-9 The integral around the closed path shown in Figure 5.1 is Sfk(s) ds fk(s) ds + -R -R (5-10) fk(s) ds From Reference (12) it is shown that if f(s) 0 uniformly as R -* then lim f f(s) eis ds = 0, R -o C- R (4 > 0) Therefore the first integral on the right in Equation (5-10) goes to zero as R c and c fk(s) ds fk(s) ds = 27 i Res(sk) i (ak+i bk) 00 = / = 27 i l + 27- i -e d(ak+i bk)] i (ak+ibk) + 27Ti -e e -ic 9 k-bk P+i ak [ -bk(d+)+i ak(d+k ) H(0) e H(d+ (P) (5-11) where bk > 0 H = Heaviside step function. Similarly, when bl < 0 the integration path is in the lower half plane and the result is F i -bbk (+i ak H(-) -bk(d+p)+i ak(d+P) F = ic 9 "ke H(-p) e H[-(d+0)] , bk < 0 (5-12) Therefore (+d>0 46 The general expression for Fk can be written in the form Fk = -i sgn(bk) c9 ak e ak H[sgn(b -bk(d+P)+i ak(d+P) -e where H is the Heaviside step function and sgn(bk) k) H[sgn(bk)(d+P)]l bk >0 bk<0 The rotation is given by Equation (5-7). 4 q = Fk() qo k=l Inverse Transformation of the Radial Deflection The radial deflection is obtained by inverting W to give isd s-e s c' s2 + c' 7 8 eis ds D(s) (5-15) The term (c' s + c' )/D(s) can be expanded in partial fractions to give 7 8 isd 1-e s 2 c sk + c k7 e 4 S(sk sm m=l "-k) - k eis ds (5-16) k m c' 7 c - 7 c 4 c' 8 c - 8 c 4 *See Appendix E for this expansion. (5-13) (5-14) 00 27 i f Wqo q% where (5-17) 00 _ 1f 27r i f Defining w Fk(0) - k 2r i isd s(s sk e (5-18) The deflection is then given by the sum = VFk) (5- qo k=l By contour integration in the upper and lower half planes, the integral in Equation (5-18) can be evaluated. Breaking the integral into two parts gives 19) (5-20) w k e i- ds ei( +d)s F ( ds- ds wk 2 [ i s(s sk) s(s sk The contour shown in Figure 5.2 is used to evaluate these integrals for 4 > 0, bk >0. Im(s) C R C s -R p R Re(s) Figure 5.2. Integration Contour for Radial Deflection for 4 > 0, bk > 0 From Cauchy's integral theorem Sw k(s) ds = /wfk(s)ds + C- R + fk(S) ds P -p -R wfk(s)ds + /wfk(s) ds C p = 2n i Res 48 From Jordan's Lemma (12) it can be seen that the integral on contour C- -*0 as -~ i.e., e lim i ds 0, > 0 S s(s sk) R-oo C~ and e i(p+d)s lim J id)s ds 0, O+d > 0 s(s Sk) R-- C- Also lim f fk(s) ds + f(s) ds P fk(s) ds 0 R- -T P -00 R-o - which is the Cauchy principal value of the improper integral. If the integral exists, then this is the correct value for the integral and the symbol P can be dropped. Therefore 00 -0J es P ds + lim e ( -s ds = 27r i Res(sk), ( > 0 (5-21) s(s s ) s(s s ) k' -0 C P The second integral in Equation (5-21) can be evaluated as lim ds = ai Res(O) p- s(s Sk) p-0 C k P Since the integration is clockwise Cf = -T7r and ( eips \ = Res(O) = ks= -1 SSk)s=0 k lim ds O s(s sk) p-O Ck Integral (5-21) now may be written a0 - 00 is e s ds s(s sk) where 0 >0 bk > 0 Similarly f ei(~I+d)s - s(s Sk) i(+d) Tri 2r i ei(Od) ds + sk sk (5-23) where p+d > 0 bk > 0 Now the case is investigated where bk > 0, 0 < 0. The contour shown below is used for this case. Im(s) ,_T__ sk ; Re(s) -~a R k C- Figure 5.3. Integration Contour for Evaluating Radial Deflection for p < 0, bk >0 7Ti Sk 7T Sk i sk + 27ri e sk (5-22) The integrals have the values S ds + lim ds =0, <0 s( k) p-0 (-k) p or e ds = i Res(O) = i < 0 (5-24) Ss(s sk) sk -m bk > 0 Similarly Ses ds = (p +d < 0 (5-25) Ss(s sk) sk o bk > 0 Equations (5-22), (5-23), (5-24), and (5-25) can be combined to give the solution for wFk(() when bk > 0 as wFk() = k i + e i H(P) + -i H(-) w 27ri sk s sk / o (~,,+d)S \ ]1 7-r + 2sir e H(+d) + H(-P-d) k sk k bk > 0 or finally Ik 1 isk 1 w k + e H((P) + H(- P) w Fk(P) s -k S[( + e ( )sk H(+d) + H(-P-d) (5-26) where bk > 0. 51 The case where bk < 0 is now investigated. Integrating in the lower half plane it is found that ds s(s sk) ei s i ds s(s sk) eiS eSk ds = 27i - s(s sk) sk bk < 0 bk < 0 7T1 + i Sk ifsk = -27ri - sk bk <0 It is evident, therefore, that the general expression for w Fk(() can be written in the form sgn(bk) Ok s k i(fsk ) H[-sgn(bk) 0] + e k H[sgn(bk) ] + 2 /2 [ 1 i(0+d)S k] 1 +e M )skH[sgn(bk)(+d)] + H[-sgn(bk)(-^)] (5-27) Inverse Transformation of the Axial Deflection The axial deflection in the s-plane can be written as isd = c' 1 ei) 5 (s) isd (1 e ) 6 2 (s s D(s) From Equation (5-1) it can be seen that the first term in Equation (5-28) can be written in terms of . Thus isd (1- e ) eis d s !( e ds 2 s D(s) (5-29) - 7 CO --0 q (5-28) c 5 c' e w Fk() 1 o S27 0 The integral in Equation (5-29) can be written as 4 Sisd ak d k (1 e ) is s k 2 e --sk ds s k= k-1 where c, 6 C = - 6 C 4 Now defining C6 k 27 00 -00 c 5 9 (0) + 9 9O isd i(s (1 -e ) e ds s'(s sk) s2(s s ) 4 uFk(Z) k=l The integrand in Equation (5-30) can be written in two parts as eis e -ds - s2(s s k) 0 ei(o+d)s J/ e ds - s 2(s -s) The contours shown in Figures 5.2 and 5.3 are used to evaluate the integrals in Equation (5-32) for the case bk > 0. Define ufk(s) uk S ei s s2(s sk) ei(+d)s s (s sk) By a procedure completely analogous to that just described for the radial deflec- tion, the results can be written j e ds = 7riRes(O) + 27riRes[ fk(sk)], -0 s (s-k) S> 0 bk >0 (5-33) 00 - 00 F (( ) uk then (5-30) Fk () k "6 k 27r (5-31) -o (5-32) uf (s) ei(P+d)s ds s (s sk) = 7riRes'(0) + 27riRes[ f(sk)] , -bk ( +i ak e (ak + i bk)2 Res[ uf(sk) u k k -bk(P+d) +i ak( -+d) e (ak + i bk)2 To find the residue of the functions at s = 0 they are expanded in a Laurent series about the point s = 0. eics e 2 s 1 s sk I + +l 2 s s 1 sk 2 3 2! 3! +- sk 2 + s +5 2 sk 1 i (P+s k + k s es2s s (s sk) 1 + 1 2 sk +i s k 2 2! The residues can be evaluated as Res(0) 1 2 s k 1 2 sk k (5-37) S i +d sk 00 -00 where O+d > 0 bk >0 (5-34) (5-35) (5-36) Res[ufk(sk)] - Sk (5-38) Res'(O) = eis e ds = - s (s s ) k k _00 00 / -CO 77i Sk Sk eisk S k 2/ k - + i(P+d)] + sLk J Integrating in the lower half plane for ^ < 0, Equation (5-30) can now be written in the form F (P) k ck sk + i ) H( ) + i( +d)) H(O+d) 1 +1 +i sk H( + ( s H(-P) I + s k - k ) H(-O-d) , 2 s k bk >0 (5-39) Performing the integration for bk < 0, the final result can be written as sgn(bk) ic ak 2 sk 1 2 (1 +i sk) +e k SH[sgn(b) + ( + ) H[-sgn(b x H[sgn(bk)p + *(l+i 0 sk) H[-sgn(bk)P] i(+d)k H[sgn(bk)(P+d)] + [1 +i(+d)sk] H[-sgn(bk)(0+d)] 2 so that ei(S+d)s s S ds s2(s s ) k bk > S> i(4+d)sk e 2 sk bk > 0 d +0 >0 Fk( ) -- [1+i(P+d)sk] + e (5-40) 1 1k 2 sk The solutions given by Equations (5-7), (5-19), and (5-31) with the correspond- ing Fk functions (5-13), (5-27), and (5-40) are substituted into the governing equations of motion in Appendix G to show that the solutions satisfy the differen- tial equations. Summary of Deflection Expressions The following expressions summarize the solution for the deflections. 4 = F k(F ) k=l 4 = wFk( ) k=l k-- 1 c 5 (qS) + c q 9 0 (5-41a) (5-41b) 4 Su k(P) k=l (5-41c) where I isk P is k(0+d) Fk( ) = -sgn(bk) ic9 ak e k H[sgn(bk)P] e sk H[sgn(bk)(P+d)] (5-41d) sgn(bk) f3k wFk( s Sk - + es H[sgn(bkl + H[-sgn(bk)p] [- +e sk H[sgn(bk)(P+d)] + H[-sgn(bk)(P+d)] (5-41e) ( ) qo sgn(bk) i c6 ak u Fk( 2 sk sgn(bk) H(y) r 1 i0sk -(1 + isk) + e [sgb H[sgn(b + (1 + i sk ) H[-sgn(bk) 1 i Sk(d) ) - -2 [1 + i sk(+d)] + e H[sgn(bk)(P+d)] +1 [1 +isk(l+d) H[-sgn(bk)(P+d)] (5-41f) bk > bk < y< y> 1 C7 sk + k 4 k 4 k k m (sk- sm) R (sk- sm) m=l m=l Sk= ak + i bk (k = 1, ..., 4) are the roots of the characteristic equation D(s) = Cs4 + i c s3 +C 2+ic s+c = 0 4 3 2 1 0 and the coefficients are defined as c fl e4 2 c = e (fe ) O 380 1 c = -EXe f 1 38 c = f (e c- e ) +e f (f rX2) 2 70 1 2 38 2 c = -Eff c 3 7 c = (f rX2)f c 4 3 7 c = f (e + e ) 5 73 1 c = ee 6 1 3 c = f7(f + eX2) 7 71 c' = e f 8 38 c' = ef +c' 9 17 8 e = E +Pr f = 1+F o o o o e = E -P f = 1+F-D e = E -P+2(G+ Pr) f = F-Pr 1 0 2 0 e = G+Pr f = G+F 3 O 3 e = I -r f = Dr +If 4 0 1 4 11 00 e = e f -e e f = r f +rf 5 01 12 5 12 0 2 e ef -e f = eD -If 6 00 1 6 4 1 01 f = D I2 7 1 0 2 f = f -rX2 8 0 1 (5-42) The derivatives of the deflections are 4 __ = E Fk, k-=l 4 q = jl w y Fi(0) k=l 4 S= uFk() +) (5-43) c 9 q 0 u k=1 where F'k() = i sk Fk()= lk esk +c k( k kc Fk() F9 k 9 C6 ak C F'() wFk(0) F, () (5-44) uk c sk +c 7 8 0, O+d / 0 Solution for No External Damping For the case of no external damping, the solutions as given in Equation (5-41) are not directly applicable. It will be noticed that the forms of the solutions are dependent upon the signs of the imaginary parts of the complex roots. Figures 4.2 through 4.8 show that there are regions in which there are only real roots of the Characteristic Equation with no damping, thus causing a problem of non- uniqueness of the solutions. Following the method of Achenbach and Sun (13) the undamped solution will be obtained uniquely by assuming the undamped solution is the limit of the damped solution as the damping approaches zero. In this manner the sgn functions in the Fk functions for Equation (5-41) can be determined. Figure 5.4 provides an example of the behavior of a set of roots as the damping, E, approaches zero. This figure shows the type of the four roots (sij = root i with damp- ing e ) for heavy damping to be two complex and two imaginary, root 1 being very near the origin. As the damping is decreased, the imaginary roots approach one another and finally meet and separate which gives four complex roots. Meanwhile the other complex roots are also approaching the real axis. This establishes the correct sign for the imaginary part of each root in the limit as e 0 and the roots all approach the real axis. Form of the Radial Deflection in Region IV The roots are all real, having the form a a From Equation (5-41b), 1 2 the radial deflection expression (when the proper signs are established) in Region IV becomes W = 2 + [H(+d) H()] qo a a a a aP1 [a le- H(-@-d) +0 F-ia -ia (0+d) H - e H(-O) e 1 H(-0-d) a 4 ~ -i a 20 -ia (+d) d) (5-45) 3P ia ia ( +d) + e H( ) -e H( 4) - e H(0) e H(04d) (5-45) s.. = Root i for External Damping External Damping c. e - 0 1 E = 2 E 3 0 0.1 1.0 10.0 Figure 5.4. Loci of the Roots of the Characteristic Equation as the Damping Approaches Zero Expanding the i3. coefficients gives P/ = -/P 1 = -~ P3 -f4 Sa + c 7 1 e 2a (a2 a) c a2 +c S-72 7 2 8 2a (a a2) 2 1 2 Substituting expressions (5-46) into (5-45) gives 2 c a + c 7 1 8 2a (a a2) 11 2 e +e 1 ) H(-0) i a,(j+d) + e -i al( +d) H(-1-d) Sia (+d) -i a (0 ) 2 c a +c ia 6 -ia \ 72 8 2 +e2 HU 2a (a -a a / 21 2 e + e H(C+d) c S 2 a a 1 2 c - 2[H(0+d) H(O)] a a 1 2 2 C a +c 7 1 e S(a a 1 1 2 X [cos (a ) H(-() cos [a (o+d)] H(-p-d)] C a + 7 2 -e c (a 0) H() cos [a (0+d)] H(0+d) 21 2 W qo (5-46) W qo (5-47) 62 Therefore, for the three distinct regions of the cylinder, the solutions are: Solution behind the load (P < -d) 2 +c - 1 cos (a ) )- cos [a (P+d)] (5-48) o a (a1 a 11 2 Solution under load (-d < 0 < 0) 2 2 c c a +c c a +c W 8 71 8 7 2 8 2 2 cos (a ) + 2 2 cos [a (P+d)] (5-49) Sa a2 a (a a2) a (a a2 2 12 11 2 21 2 Solution ahead of load (0 > 0) cW 2 + c 7 72 2 cos (a4) cos [a ( }+d)] (5-50) a2 (a2 a2 2 2 o a(a a2) 21 2 Form of the Radial Deflection in Region VII As in Region IV, the roots are a a However, there can be no deflection 1 2 ahead of the load in this region because the load speed is greater than any of the wave speeds in the material. This zero displacement comes about mathematically because the roots all approach the real axis from the negative imaginary direction. The solution for this region has the form S2 2 2 2 2 o a a a(a- ) 12 11 2 x {cos (a4 )H(-p) cos [a (O+d) ]H(-p-d) c a +c 2 \ + 7 2 2e 'cos (a2 )H(- ) cos [a (0+d)]H(=P-d) (5-51) 2 2 2 2 a (a a ) 21 2 Form of the Radial Deflection in Region V The roots in this region have the form a, ib. The radial deflection in this region is given by the expression W c q [H(-Ip) H(-P-d)] qo a2 b2 2 2e cos(ao)H(-p)-cos [a(+d)] H(- -d) a (a + b ) c b2 +c + [e-b H(P) ebO H(-P) e-b(I+d) H(P+d) 2b2 (a + b ) + eb(d) H(--d)] (5-52) Form of the Radial Deflection in Region VI In this region the roots are all imaginary, of the form +i b i b This 1 2 gives an exponentially decaying solution as in Regions I and III. The radial de- flection expression is given below. W c S[H(M+d) H(4)] qo b b 1 2 c c b -b b -b ((+d) + 271 e H( ) e H(- ) e H( +d) 2b (b b ) 1 1 2 b (+d) + eb H(-P-d)] c c b2 b -b2 b ((+d) + 8 7 2 [e2 H(-O) e 2H() e 2 H(-O-d) 2b(b b) + e d)H(4+d) (5-53) Comparison of Solution with Other Results As a comparison of the results of this analysis with another theory, the static problem of a distributed pressure load on an isotropic shell was considered as shown in Figure 5.5. Figure 5.5 Static Load Problem For the static problem shown above the roots of the characteristic equation, excluding prestress, are complex. The solution for the region under the load as given by the present theory can be reduced to a L K 2 eaL 2 1 K. 2 < -aL 1 cos aL e 2 cos a L) + -a L + K 2 e sin a L + e 2 -a L sin aL ) L - 2 R' where (5-54) L - 1 R 65 This problem is solved by Timoshenko (14) and the deflection given by his theory, when put in a compatible form, becomes eW -2 -/ 1 = K' 2 -e cos P3 e cos L (5-55) The second term appearing in Equation (5-54) is missing from Equation (5-55). This additional term arises because of the inclusion of shear deflection which was not present in the Timoshenko theory. A comparison of the results of these deflection expressions is made in Figure 5.7. Numerical Results A computer program was developed for the calculation of the displacements and stresses determined in this research. The general expressions given by Equations (5-41) were programmed for the displacement solutions and the stress calculations are discussed in Chapter VI. A flow diagram of the computer pro- gram is shown in Figure 5.6. The details of the program can be found in Ap- pendix H. It is written in Fortran for time share computer application. Comparison of Results with Other Solutions for a Static Load The radial deflection for a static distributed load on a cylindrical shell is given in Reference (14). As a check on the solution this static problem was solved using the present results and the comparison is shown in Figure 5.7. The results agree very well. The effect of variation in the thickness-to-radius is also illustrated in Figure 5.7. The rotation and axial deflection are shown for this static problem in Figure 5.8. In addition to showing the form of the displacements for the static load, Figures 5.7 and 5.8 serve as a basis against which the dynamic displacements can be compared. The deflections are sym- metric about 4 = -0.5 for the static load. Initiate Program Figure 5.6. Flow Diagram for Computer Program for Deflection and Stress Calculations S .I iA1L1 111 ilhAIT . I .I I I I, I 4 4 4 4 4 4 4 4 44 1 Jr L Figure 5.7. Radial Deflection for Static Load on an Isotropic Shell (p = 0.3) -1.0 Figure 5.8. -I--- Displacements for a Static Load on an Isotropic Shell (p= 0.3, h/R= 0.01) 0.15 0.10 0.05 U/q 0 -0.05 69 Summary of Deflection Response for Shells under Various Load Velocities A summary of some of the types of deflection patterns assumed by a shell for increasing load speed is shown in Figure 5.9. For the particular properties used for this example, the various regions (root types) associated with each waveform can be found by inspection of Figures 4.2 through 4.8. For example, for no damping, positive prestress corresponding to internal pressure, h/R = 0.001 and the material properties given in Figure 5.9 (properties are those corresponding to an isotropic shell as shown in Appendix I), Figure 4.2 can be used to associate load speed with root type. Following the vertical line of h/R = 0.001, it is evident that X2 = 1 lies in Region I where the roots are all imaginary. This gives a critically damped exponentially decaying solution as shown by Equation (5-53) and is shown in Figure 5.9(a). As the load speed increases Region II is entered where the roots are complex. This is the form of the static load problem roots, and if no pre- stress existed Region II would extend from zero load speed up to the first criti- cal, which is at X The solution for 2 = 2 is shown in Figure 5.9(b), and is exponentially decaying. The response becomes sinusoidal after crossing X = X At a load speed just greater than 2 the deflection response has a very short period. A small amplitude wave train precedes the load and a large amplitude wave follows it. As the load speed increases the sine wave period increases as shown in Fig- 2 ure 5.9(d) for X = 30. These sinusoidal deflection patterns are in Region IV where the roots of the Characteristic Equation are all real. The mathematical expression for W/qo is given by Equation (5-47). Crossing 3 into Region V, the roots are real and imaginary. Equation (5-52) gives the radial deflection, and 2 Figure 5.9(e) shows the response to be a long period sine function for A = 500. h R E0 E,,o Ex Ex0 0.001 1.0 = 0.35 = 0.30 = 0.004 = 0.002 = 1.0 = 0 Figure 5.9. Radial Deflection Shape for Various Load Velocities Jumping to Region VII brings a longer period sinusoidal oscillation as illus- trated by Figure 5.9(f) and the response in this region, where the roots are again all real, was discussed previously. The radial deflection is given by Equation (5-51). Because Region VI covers such a limited range in velocity the response was not included in the summary but is discussed later. Region II Response A study of the response of an isotropic shell at a load speed below the first critical was made to determine the effect of external damping. These results are shown in Figure 5.10 where the damping ranges from very light to very heavy. Of course, when damping is introduced the root form is no longer the same as that of Region II. Region IV Response 2 The short period sinusoidal response of the radial deflection at X = 2.7 is shown in Figure 5.11. As the radial damping is increased this response is changed drastically as shown in Figure 5.12. The response for a damped sys- tem, which was in Region IV with e = 0, approaches closely that of the Region II 2 behavior. Figure 5.13 shows the radial response at X = 5 and 10. The maxi- mum amplitude remains constant as the period of the wave increases for greater load velocities. Damping Effect on Regions V and VII Response 2 The effect of damping on the wave forms for X = 500 and 2000 is shown in Figures 5.14 and 5.15. The amplitudes of the sinusoidal deflection response are initially decreased, and, as the damping becomes greater, the response becomes critically damped and the deflection approaches zero with an increase in distance from the load. = 0.001 = 0.01 = 0.11 = 1.0 = 0 Figure 5.10. Radial Deflection Response for Variations in Radial Damping (" = 2.0) 0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 q0o G2 S- 0.35 E xo = 0.30 I Eo I P = 0.004 1 I F = 0.002 d = 1.0 = 0 -11 -2 0 I -1 - -___________ -3 -2 -1 0 1 Figure 5. 11. Radial Deflection Pattern Immediately Above the First Critical Load Speed - d R -1.0 0 X- =2.7 I h Il- = 0.001 \ I I / Efl IE0 = 1.0 -1 Gxz = 0.35 Exo EXo 0.30 xo -2 P = 0.004 F = 0.002 d = 1.0 e = 0.01 --- = 0.10 Figure 5.12. Change in the Radial Deflection Pattern with Increasing Damping (X2 = 2.7) --- d -1.0 2 (a) X2 =10 1 -1.5 -0.5 S = 0 d = 1.0 F = 0.002 P = 0.004 h S= 0.001 R E00 S= 1.0 XZO S= 0.35 w Ex qo E" = 0.3 Ex 0 -1 \ T^ /2 Figure 5.13. Deflection Wave Form at X2 = 5 and 10 -1.0 d -1 0 w 3.0 qo 2.0 1.0 1 0 2 1 0 2 500 h 0.001 R -2.0 E00 = 1.0 EX Gxzo -3.0 = 0.35 EXO Eo -p = 0.30 xo P = 0.004 F = 0.002 d = 1.0 Figure 5.14. Effect of Damping for X = 500 d L R L CL---^ -- G XZO Exo EV -o Exo P F d 0 - 0 = 0.35 = 0.30 = 0 004 = 0.002 = 1.0 Figure 5.15. Effect of Damping for X2 = 2000 Region VI Response This region has a deflection pattern which is almost totally axial. Three critical velocities have been crossed to get into this velocity range which corre- spond to the longitudinal, shear, and bar wave speeds. Therefore, there can be no bending effect transmitted. The behavior is like an axial compression on a membrane which expands radially, as shown by Figure 5.16. The effect of prestress is observable in this figure. The maximum deflection is increased by about 20 percent when going from external hydrostatic pressure to internal hydrostatic pressure. This axial mode of deflection also appears in other velocity ranges. For instance, the broken lines in Figure 5.16 show the behavior at X = 1001. This is the range between X = 1000 and A2 = 1002 where the roots are real and imaginary. Another example is shown in Figure 5.22 where E0o/Exo is less than 0.08 and this occurs in Region VIII as shown in Figure 4.4. 2 Deflection Behavior in the Vicinity of CR The first (lowest) critical load speed occurs at iCR = = 2.552. Figure 5.17 illustrates the unbounded response of the deflection as that speed is ap- proached. The effect of damping on the maximum deflection is also illustrated. There are four other critical velocities as discussed in Chapter IV. 2 Effect of Prestress on 2 'CR The effect of the axial prestress on the location of the first critical load speed is shown in Figure 5.18. This effect is also observable in Figure 4.8, since the first critical load speed is at A2. The circumferential prestress does not have a significant effect on load speed, as shown in Figure 4.7. -1.0 0 0 0.8 h 0.001 E0 = 1.0 -.. .\ Gxz Ex- 0.35 E .qo Eo .\\O.30 Ex U --xo o -_.. _ -6.0 -4.0 -2.00 2.0 4.0 \2 = 940 P = 0.004, F =0.002 _ 2 S P=0,F=0 -22 P=-0.004, F=-0.002 2 = 1001 -4 --* P =0.004, F = 0.002 Figure 5.16. Deflection Response for Variations in Circumferential and Axial Prestress 2.5 2.552 2.6 Figure 5.17. Maximum Radial Deflection in the Vicinity of the First Critical Load Speed w max qo 81 C oI C4 -e S ________ cM ----- ------------- 1-1 0 0 oC I C - - _______ 00 IIn n c, I 0 C a on eooo co , a - ^ Superposition of Step Loads The effect on the radial deflection can be observed in Figure 5.19(a)-(f) where the load length was varied from 0.1 to 5.0. By superposing various com- binations of step loads it is possible to approximate any shape of load desired. As an example of this type of application, the radial deflection response from a symmetric sine wave type load and a sharp edged pressure front was deter- mined. The results of these calculations are presented in Figures 5.20 and 5.21, respectively. Study of Material Properties Variations A look at the effect of decreasing the E0o/Exo ratio is summarized in Fig- ure 5.22. Starting in Region IV, as can be seen in Figure 4.4, the ratio is de- creased from 1.0 (as for an isotropic material) to 0.04. As the ratio is lowered, the maximum deflection gets large rapidly, and becomes unbounded as 2= 2CR is approached. After crossing A3CR into Region VIII, the strength in the cir- cumferential direction is of course very low and the material is of little interest for engineering applications. The same type of response as in Figure 5.22 will be obtained by increasing Evo/Eo significantly. This can be observed by inspecting Figure 4.6. For other types of material property variations, the general response can be pin- pointed by observing the type of roots at the particular location through the use of Figures 4.2 through 4.8 and using the numerical results presented here show- ing similar calculations of deflections. qjOR (a) d = 0. 1 Figure 5.19(a). Variation of Pressure Pulse Length, d, at A2 = 30 A h R E0o EL Exo E0 10 P F c d 0 = 30 -0.001 1.0 = 0.35 -0.30 0.004 - 0.002 =0 - 0.1 |

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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EI378QK26_BKV331 INGEST_TIME 2017-07-17T20:08:21Z PACKAGE UF00098431_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES PAGE 1 ORTHOTROPIC CYLINDRICAL SHELLS UNDER DYNAMIC LOADING By ELMER MANGRUM, JR. A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTLU. FULFILLMENT OF THE REQUIREMENTS FOR THB DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1970 PAGE 2 UNIVERSITY OF FLORIDA 3 1262 08552 3057 PAGE 3 This dissertation is dedicated to my wife Rita and my daughter Gaila. PAGE 4 ACKNOWLEDGMENT I would like to acknowledge the support and encouragement of General William M. Thames, K. A. Campbell, and N. E. Munch of the General Electric Company who made this research possible. I wish also to express my sincere gratitude to Dr. J. J. Burns for this guidance and suggestions during the course of this research. ill PAGE 5 TABLE OF CONTENTS Page LIST OF TABLES vii LIST OF FIGURES viii KEY TO SYMBOLS xi ABSTRACT xv Chapter I INTRODUCTION 1 Statement of the Problem 1 Specific Goals of This Research 2 Review of Previous Work 3 Contributions of This Work 3 n GOVERNING EQUATIONS OF MOTION 5 General Equations 5 Axisymmetric Loading 15 Pressure Loading Form 15 Nondimensional Equations 16 m TRANSFORMATION OF EQUATIONS 20 IV INVESTIGATION OF THE CRITICAL VELOCITIES 23 IV PAGE 6 TABLE OF CONTENTS (Continued) Chapter Page V SOLUTION FOR DISPLACEMENTS 42 General Solution 42 Transformed Displacements 42 Inverse Transformation of the Rotation 43 Inverse Transformation of the Radial Deflection ... 46 Inverse Transformation of the Axial Deflection ... 51 Summary of Deflection Expressions 55 Solution for No External Damping 58 Form of the Radial Deflection in Region IV 59 Form of the Radial Deflection in Region Vn 62 Form of the Radial Deflection in Region V 63 Form of the Radial Deflection in Region VI 63 Comparison of Solution with Other Results 64 Numerical Results 65 Comparison of Results with Other Solutions for a Static Load 65 Summary of Deflection Response for Shells under Various Load Velocities 69 Region 11 Response '^1 Region IV Response 71 Damping Effect on Regions V and VII Response ... 71 Region VI Response 78 Deflection Behavior in the Vicinity of ^.^qj^ 78 Effect of Prestress on Xi^p 78 Superposition of Step Loads 82 Study of Material Properties Variations 82 v PAGE 7 TABLE OF CONTENTS (Continued) Chapter Page VI STRESSES 92 Development of Stress Equations 92 Numerical Results 94 Vn CONCLUDING REMARKS 98 Conclusions 98 Suggestions for Future Work 99 Appendix A FOURIER TRANSFORM OF THE FORCING FUNCTION .... 100 B SOLUTION OF EQUATIONS FOR THE TRANSFORMED DEFLECTIONS 102 C EVALUATION OF THE DISCRIMINANT OF A FOURTH ORDER POLYNOML\L 106 D DETERMINATION OF THE CRITICAL VELOCITY EQUATIONS Ill E COMPUTER PROGRAM FOR SOLUTION OF LOAD VELOCITIES WHICH CAUSE REPEATED ROOTS IN THE UNDAMPED CHARACTERISTIC EQUATION 116 F PARTL\L FRACTION EXPANSION OF A FOURTH ORDER POLYNOMIAL 123 G CHECK TO SEE THAT SOLUTIONS SATISFY THE GOVERNING DIFFERENTIAL EQUATIONS 125 H COMPUTER PROGRAM FOR DEFLECTION AND STRESS CALCULATIONS 130 I RELATIONSHIPS BETWEEN ANALYSIS PARAMETERS .... 147 BIBLIOGRAPHY 149 ADDITIONAL REFERENCES 151 VI PAGE 8 LIST OF TABLES Table Page 4-1 Correlation of Root Type with Region Numbers for Figures 4 . 2 through 4 . 8 27 H-1 Options Available for Program DEFSTR 131 Vll PAGE 9 LIST OF FIGURES Figure Page 2 . 1 Cylindrical Coordinate System 5 2.2 Pressure Loading 16 4.1 Flow Diagram of Computer Program VCRIT which Determines Load Velocities at which Repeated Roots Occur 28 4.2 Classification of Roots of the Undamped Characteristic Equation for Variations in the Thickness-to-Radius Ratio Including Prestress 29 4 . 3 Classification of Roots of the Undamped Characteristic Equation for Variations in the Thickness-to-Radius Ratio with No Prestress 30 4.4 Classification of Roots of the Undamped Characteristic Equation for Variations in Eq /E^ 31 4.5 Classification of Roots of the Undamped Characteristic Equation for Variations in Gxz /^x 32 4.6 Classification of Roots of the Undamped Characteristic Equation for Variations in E,, /EÂ„ 33 J^o Â•'^o 4.7 Classification of Roots of the Undamped Characteristic Equation for Variations in the Circumferential Prestress 34 4.8 Classification of Roots of the Undamped Characteristic Equation for Variations in the Axial Prestress 35 4 . 9 Path of the Roots of the Undamped Characteristic Equation for Increasing Load Speed (Beginning in Region I) 37 4.10 Path of the Roots of the Undamped Characteristic Equation for Increasing Load Speed (Beginning in Region V) 39 5.1 Contour Integration Path for Evaluating Rotation Integral ... 44 5.2 Integration Contour for Radial Deflection for > 0, b, > . . 47 5.3 Integration Contour for Evaluating Radial Deflection for 49 Vlll PAGE 10 LIST OF FIGURES (Continued) FJRure Page 5.4 Loci of the Roots of the Characteristic Equation as the Damping Approaches Zero 60 5.5 Static Load Problem 64 5 . 6 Flow Diagram for Computer Program for Deflection and Stress Calculations 66 5.7 Radial Deflection for a Static Load on an Isotropic Shell (M0.3) 67 5.8 Displacements for a Static Load on an Isotropic Shell (M 0.3, h/R = 0.01) 68 5.9 Radial Deflection Shape for Various Load Velocities 70 5.10 Radial Deflection Response for Variations in Radial Damping (A^ 2 . 0) 72 5.11 Radial Deflection Pattern Immediately Above the First Critical Load Speed 73 5 . 12 Change in the Radial Deflection Pattern with Increasing Damping (> 2 = 2.7) 74 5.13 Deflection Wave Form at A^ = 5 and 10 75 5 . 14 Effect of Damping for A^ = 500 76 5.15 Effect of Damping for A^ = 2000 77 5.16 Deflection Response for Variations in Circumferential and Axial Prestress 79 5.17 Maximum Radial Deflection in the Vicinity of the First Critical Load Speed 80 5.18 Effect of the Axial Prestress on the First Critical Load Speed . 81 5.19 Variation of Pressure Pulse Length, d, at A =30 83 5.20 Response from a Smooth Sine Wave Type Pressure Pulse Using Superposition 89 5.21 Response from a Sharp Pressure Front Using Superposition . . 90 5.22 Radial Deflection Response for Variations in the Circumferential Modulus 91 IX PAGE 11 LIST OF FIGURES (Continued) Figure Page 6 . 1 Bending Stress in an Isotropic Shell Under a Static Load ()Li = 0.3, d = 1) 96 6.2 Surface Stresses in an Isotropic Shell Under a Static Load (M = 0.3, h/R= 0.1) 97 PAGE 12 KEY TO SYMBOLS X, e, z coordinate axes K^ (fv = X, 6, z) unit vectors in coordinate directions R radius of cylinder (to the middle surface) h thickness of cylinder u, V, w displacement in directions of coordinate axes 4>, f) rotations t time N N (a , /3 = X, 0, z) stress resultants M^, M (a, /3 = X, 6, z) moment resultants shear force axial prestress stress resultant circumferential prestress stress resultant moment of inertia defined in Equation (2-7) initial lateral pressure mass density strain in a direction shear strain stress in a direction shear stress modulus in x direction modulus in e direction modulus in normal direction XI Q^(a = X, 0) PAGE 13 Gq,o (a,(i ^ X, e, z) shear moduli D {a = X, 0, p) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) defined in Equation (2-12) correction factors time varying lateral pressure load damping coefficient coefficients of a matrix Heaviside step function magnitude of lateral pressure load constant velocity constant defined in Equation (2-29) U, W dimensionless displacements in axial and normal directions
the transforms of derivatives of f((/)) are given by {\s) = (is)^f(s) (3-3) so I dcf.^ J is f (s) (3-4) F ^^ -s^f(s) (3-5) 20 + -2 +e H[sgn(bj^)0] +2 H[-sgn(bj^)0] ^ is,((^+d)' H[sgn(bj^)((^+d)] + 1 H[-sgn(bj^)(<^+tl)] (5-41e)
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