7D-A157 637 ON THE NOTIO N OF RODS WITH INITIAL SPACE CURYATURE(U) 1/1ARMY BALLISTIC RESEARCH LAB ABERDEEN PROVING GROUND NDH B KINGSBURY JUN 85 BRL-TR-2658

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US ARMYMATERIEL

COMMAND TECHNICAL REPORT BRL-TR-2658

In t

ON THE MOTION OF RODS WITH INITIALSPACE CURVATURE

Herbert B. Kingsbury

June 1985

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ON THE MOTION OF RODS WITH INITIAL SPACECURVATURE Final Report

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* University of Delaware

KEY WORDS (Continue on revere, side it neceeary aid Identify by block number)

Tube Flexure:. Curved Gun Tubes,-Gun Tube Vibrations, Imperfections in Gun Tubes;Elasticity.( --

20. AbI kACr (Cmtiuae ass- eav tlib if neeeary ad Identify by block number)

- Previous formulations and models of the vibrations of gun tubes havebeen based on the assumption of an axisymmetric distribution of material, i.e.a perfectly made albeit curved gun tube. This report contains a newformulation for the mathematical problem related to gun tube vibrations withthis constraint removed: the formulation admits non-colinear centroidal andbore axes, reflecting the actual case with prpduction gun tubes. Thesefactors may play a significant role in "Jump" inherent in direct fire weapons - ,

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20. (continued)

4 including aspects related to fleet zerox4r other zeroing procedures in tankguns.

The general large deformation equations of motion are derived for alinearly elastic rod with initial space curvature. The equations allowexamination of such effects as transverse shear deformation, rotatory inertia,initial twist, variable cross section size and shape, and variable curvatureand torsion on the motion and deformation of the rod. A linearizedengineering theory is developed from the general results and equations forseveral special cases re fomulatA-dh -The detailed mathematical approach isdiscussed. V--r--,ovc -

The study includes only theoretical aspects and derivations related tothe formulation. Studies to investigate the importance of the newly-introduced terms on gun tube vibrations and flexure, and the numericalimplementation of the formulation have not yet been undertaken. Additionalwork is planned.

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TABLE Or COENTS

Page

LIST OF ILLUSTRATIONS ........................................ 5

I. INTRODUCTION....... ................................... .. ... *. 7

II. EQUILIBRIUM EQUATIONS FOR A ROD ELEMENT....................... 9

A. Force and Moment Equilibrium .............................. 9

B. Inertia Forces and Couples. ............. ............... . 21

III. STRAIN-DISPLACEMENT EQUATIONS ...... ...................... 23

IV. A TECHNICAL THEORY OF RODS WITH INITIAL CURVATURE AND TORSION.. 29

A. Displacement Functions .... ...... .......... o... .......... 29

B. Strain-Displacement Equations ... ..... 0 ........... ..... 30

C. Inertia Forces and Moments ... ..... ..... ... ..... . ... ...... 33

D. Moment-Curvature And Force-Displacement Equations .......... 34

E. Displacement Equations Of Equilibrium Neglecting TransverseShear Deformation And Rotatory Inertia .................. 37

1. General Form ...................... ................... 37

2. Straight Rod With Variable Section Properties ...... o... 39

3. Rod Curved In A Plane With Constant Curvature AndVariable Section Properties ............................ 40

4. Rod Curved In A Plane With Variable Curvature ButConstant Section Properties ......... ................... 41

5. Rod With Constant Curvature And Torsion And WithVariable Cross Section .......... ... ...... .............. 42

T. CONCLUSIONS ....................so............. . . . ............ 44

ACKNOWLEDGMENT ............................. ... ............... 44

REFERENCESo... .... . . .. . . .. . . .. . . . ...... .... 45

BIBLIOGRAPHY ............... ... .......... o . . ........... o .. . 47

APPENDIX A - SPACE CURVE GEOMETRYo ..... .... oo......... 51

1. Vector Tangents Of A Space Curve (T) ...................... 52

3

TAJi O CoOrUTKS (coutinued)

Page

2. Vector Binormal Of A Space Curve (B) ..................... 54

3. The Vector Principal Normal Of A Space Curve (v) ........ 56

4. The Vector Radius Of Curvature Of A Space Curve( p) ..................................................... 56

5. The Radius Of Curvature...........so.............. .. ... 58

6. The Serret-Frenet Formulae.. ... . ..... ... ..... * .... . ... .. 59

7. Example ................................................. 62

APPENDIX B - COEFFICIENTS APPEARING IN EQUATIONS (IV-ll)... 69

DISTRIBUTION LIST ........... .............. . ..... ..... .... 77

4

LIST (1 ILLUSTRATIOS

Figure Page

1 Space Curve Coordinates.................................... 10

2 Stress Sign Onvention ....... .. ........................... 11

3 Force and Moment Resultants On Rod Element .................. 13

4 Free Body Diagram of Rod Element........................... 15

5 Adjacent Cross Sections Before Deformation.................. 24

6 Displacements of Points of Adjacent Cross Sections.......... 25

7 Displacement and Rotation Components ........................ 31

s

I. IETEOUCTIOI

The work described in this report was motivated by the need to explainobserved inconsistencies between measured dynamic response data and responsepredictions of various analytical and finite element models of certain rod-like structures.

It was felt that a sufficiently general mathematical model of this classof structures might illuminate hitherto unnoticed dynamic effects andinteractions caused by such phenomena as large displacements and rotations,initial space curvature, variable shape and area of cross section, initialtwist of cross section principal coordinates along the rod axis, transverseshear deformation and rotatory inertia.

Although the study of vibrations of rods and bars is one of the mostancient in structural mechanics, a completely general formulation of theproblem is difficult to find in the literature.

One of the seminal treatments of the derivation of equations of motion ofrods with initial curvature appears in the text on Elasticity by A.E.H.Love.1 Although restrictive forms of Love's equations are used directly, orare re-derived, by subsequent investigators of small amplitude vibrations ofcurved rods, there are deficiencies in Love's work which render his strain-displacement equations unsuitable as the starting point for a generalexamination of the motion of rods with space curvature.

In Love's work, the final state of deformation is assumed to be such thatcross sections remain plane and normal to the centerline of the deformedrod. By these assumptions not only is transverse shear deformation excludedbut, as will be shown, kinematical inconsistencies are introduced.

Love employs a two stage deformation process consisting of first theimposition of twist and curvature nn the initially straight rod centerlinefollowed by imposition of small displacements to points of the crosssection. This approach leads to confusion of local coordinate systems and ofdisplacement components which renders suspect his expression for the change inlength of line segment connecting adjacent points. The generality of thestrain-displacement equations presented is further limited by the assumptionsthat displacements are small and that dimensions of the rod cross section aresmall in comparison to the radius of curvature. Finally, strain-displacementequations for an initially curved rod appear explicitly only in a restrictive,plane curvature, form.

Equations governing both static and dynamic deformation of curved rodsare employed in such areas of structural mechanics as vibrations of curvedbeams, large deflection and stability analysis of beams and columns and non-linear dynamics, including stability of oscillations of beams and strings. Abibliography of this literature with emphasis of curved beam vibrations isincluded with this report.

1 A.E.H. Love, A Treatise On The MAthematical Theory of Elastiity, 4th Ed.,

Dover PubZication, New York, 192?.

7

In spite of the large number of papers dealing with these subjects, theredoes not appear to have been a re-examination of Love's strain-displacementequations or a more general independent derivation. The equations of motionemployed by the various investigators are either those presented by Love orare derived for each particular application on an "ad-hoc" basis.

In this report a derivation of the equilibrium equations for a rod withgeneral space curvature is presented in Section II. This derivation issimilar to that presented by Love, or more recently by Kachanov. although itis somewhat more rigorous than those in its examination of the relativeimportance of higher order terms in Taylor's series expansions.

A completely general set of small strain but large displacement androtation strain-displacement equations is then formulated in Section III for arod with initial twist and arbitrary space curvature. In these equations thedisplacement components are referred to the local tangent, principal normaland bi-normal space curve coordinates rather than to fixed globalcoordinates. No assumptions regarding the structural action or mode ofdeformation of the rod are made so that the functional form of thedisplacement components is left unspecified.

In Section IV a linearized form of the general strain-displacementequations is first presented. A technical theory of rods with space curvatureis then developed based in the usual Leam theory assumptions that crosssections remain plane and undeformed. The displacements are then expressed asappropriate linear combinations of three central curve displacement componentsand three rotation components.

Based upon these displacement functions, the inertia force and momentterms appearing in the equilibrium equations can then be formulated in termsof time derivatives of these six displacement variables.

The linearized strain displacement equations phrased in terms of thecentral curve displacements and cross section rotations are then used toderive force-moment-strain-curvature equations for a linear elastic rod. Theequations are then further simplified by elimination of transverse shear androtatory inertia effects. When these equations are combined the equilibriumequations, a system of four coupled equilibrium equations in the fourdisplacement and rotation variables results.

The final section of the report is devoted to examination of thedisplacement equilibrium equations for special cases in order to examine theconditions under which various uncouplings of motions occur.

2 L.M. Kachanov, "A Brief Course On The Theory of Buckling" TAM Report No.441. Univereity of Illinois Urbana-Champaign, 1980.

8

II. EQUILIBRIUM EQUATIONS TOR A RD EUMKUT

A. Force and Moment quilibrium

A typical element of a rod with space curvature is shown in Figure 1.

The line joining centroids of cross sections is a space curve whoseradius (p), cr curvature K-l/p, and torsion (W) *are arbitrary functions ofarc length (a). The shape of the cross sections is arbitrary but assumed tobe a continuous function of s. At the centroid of any cross section anorthogonal coordinate system (x,y,z) is constructed with corresponding unitvectors i, j, k, such that k is the unit tangent vector to the space curvedetermined by the line of centroids. Vector k therefore points in thedirection of increasing s. Unit vectors i and j are the principal normal andbi-normal vectors respectively so that jxk - i.

The principal directions of the cross section area of the rod withrespect to the centroid are, in general, inclined with respect to the x, y

coordinates. The angle between these two sets of coordinate axes may vary

continuously with a for the case of a pre-twisted rod.

Figure 2 shows a section of curved rod of incremental length 6s lyingbetween points 0 and 0' of the space curve of centroids. By the usualdefinition, the face at 0 is a negative face since its outward normal pointsin the direction opposite to the local tangent vector while the face at 0' isa positive face. The usual sign convention for stress components in positive

and negative faces will be employed.

The force and moment resultants of the stress components (axz, yz, a zz)acting on a cross section of current area A are defined as

V = 1 0 dA , (la)**

V - f o dA , (lb)y fA yz '

V a f a dA , (lc)

?KAy a dA , (2a)

M - - fA x adA , and (2b)y A z

Appendix A presents a di sussion of space curve geometry

Equations are numbered consecutively within each section.

9

x xg

00,~C -4i/

~1y

Figure 1. Space Oirve 0:ordinates

10

Figure~~~~~ 2.Srs in bvni

.;..2....i....

Cy Q1 UIQ ,

f- 40U O UQ Q

y

Figure 6. Displacements of Points of Adjacent Cross Sections

25

Co

Figure 5. Adjacent Cross Sections Before Defornation

24f

m I f {y F - x FiyI dA f A yyu dA - A yxv dA . (26c)A A A

II. STRAI-DISPIAKKNJT EDUQTIONS

Equations relating the components of strain at a point in the rod to the

displacement field are derived by considering the change in length of a linesegment connecting to arbitrarily close points of the rod as the rod is

deformed.

Figure 5 shows two adjacent cross-sections of the rod before

deformation. The centroidal curve has initial curvature K and torsion A0 0

The centroid of one cross-section is point 0 and the other, at a small

distance 6s along the curve, is 0'. The vector principal normal, binormal and

tangent at 0 are i, j and k respectively while the corresponding vectors

at O' are denoted by i, j and k. Coordinates x, y, z directed along i, j andk respectively comprise a local coordinate system with origin at 0.

Corresponding coordinates at 0'are x, y and z.

Point Q lies in the cross-section at 0 while Q' lies in that at 0'. It

is assumed that Q and Q' are initially arbitrarily close so that the position

vector of Q', r can be expressed as

r = 6x i + 6y j + 6z k (I)

where 6x, 6y, and 6z are arbitrarily small distances.

If n is a unit vector parallel to r and directed from Q to Q', and r is

the magnitude of r . Then

r - rn - r X i + rm j + r n k (2)

when £,m, and n are the direction cosines of n relation to i, j and k

respectively. Equations (1) and (2) imply the relationships

6x - rX, 6y = rm, and 6z = rn . (3)

The length of the position vector joining material points initially at Q

and Q' as the rod undergoes a deformation which carries these points

to Q and Q' respectively is next determined.

23

llill i ~i! illlll i ............. .. .....

For a discussion of various representations of the angular acceleration andang~lar velocity vectors the reader is referred to a recent book by Kane, et.al.

if

7 1 iF Ix i + F Iy + F Izk

the scalar inertia forces become

Fi = - YU, (24a)

F -y=- yv and (24b)

Fi = - TV. (24c)

Upon integrating over the area of the cross section, A, the inertia forces perunit length of rod arising from the three body force components are nextobtained,

T IX =f A F IdA - - fA-y udA , (25a)

T - f A F IydA - - f y vdA , and (25b)

T1 z = 1fA F IzdA -- fJy wdA. (25c)

Finally the inertia force couples are presented in terms of the threeacceleration components

m x f y F IdA A y ywvdA, (26a)

*Iy - f x F IzdA - f A yxwdA , and (26b)

4T.R. Kane, P.W. Likina, and D.A. Levinson0f, "Spacecraft Dynamics," McGrawHillZ, New York, 1983.

22

The reader is referred to a paper by E. Reissner3 for corresponding largedisplacement curved beam equilibrium equations in which the force and momentcomponents are referred to the undeformed coordinates.

B. Ineztia Forces and Couples

The inertia body force, FI, acting at any point, Q, in a cross section isdefined by

S- Y U Q (22)

where y is the mass density of the material of that point, u is thedisplacement vector associated with that point and superscript "dot" indicatesdifferentiation with respect to time.

The displacement vector uQ is expressed in terms of the local space curvecoordinates as

uQ-u i+v j +wk

where u, v, and w are the scalar displacement components in the x, y and zdirections respectively.

Since the local coordinate system is moving in the fixed (inertial)reference frame, the acceleration vector may be expressed in the form:

UQ- aQ + a x uQ+ x )+ 2 x vQ (23)

where

v- ini+vJ k

aQ-u i+vj +wk

and i and a are the angular velocity and angular acceleration vectorsrespectively of the local coordinates in the fixed reference frame. Forproblems involving small rotations and/or small displacements of the rod crossthe non-linear terms in Eq. (23) may often be neglected in comparison with Q.

Since this report will be concerned primarily with equations governingsmall motions the non-linear terms in Eq. (23) will henceforth be neglected.

E. Reisener, "On One-Dimensional Large-Displacement Finite-Strain BeamTheory," Studies in Applied Math, Vol 52, pp 87-95, 1973.

21

V(s + ) V + + V j + V k andx y z

u(s) m x I + m j + M k

Eq. (19) becomes

aM-Xm + K M -V + mi

y z y x

aM+ ( -Y + XM + V + m )j

38 x x y

aM+ (a-s - M + m )k = 0x z

The three scalar equations representing the conditions of moment equilibriumfor the rod element become

aMX- XM + K M - V + m = 0, (21a)as y z y x

3M+ )M + V + M = 0 , and (21b)38 x x y

aMz _ KM + m 0. (21c)

3s x z

Although this manner of derivation of the equations of force and momentequilibrium for a curved rod used above is different from that presented byLove the resulting equations are essentially identical. Differences in formresult from differences in force and moment sign conventions and because Loveemploys local coordinate axes which are the principal directions of the areaof the local cross section of the rod rather than the natural space curvecoordinates.

In general, the local x, y, z, coordinates are the deformed coordinatesof the centroidal curve of the rod. The torsion and curvature terms appearingin Eqs. (12) and (21) are the total values of these quantities consisting ofthe sums of their initial values and their respective changes due todeformation of the centroidal curve. For small deformation analysis theinitial coordinates and initial values of X and K are employed.

20

I'z V (S + 62) - [k60 + K/2 i (6s) 2 x [V (a+ ) + L)l +6s

+ -2v a 2 (68)2]s2 s

kxV (s + ) 6s + [1/2K i x V(S + ) + k x V 1 +] (68)2 and (17)as

asWo. + 6s) -M(s+) + 6s + 1/2 L2}/ 1 +(6)2 (18)

s 1 s 2 a

where again V(s+ ) and N(s+ ) indicate the force and moment resultants acting

on the positive face at 69 - 0 so that M(s+ ) = - m(s).

Finally, substitution of the result given by Eqs. (14) and (16) through(18) into Eq. (13) and division by 6s yields the following vector momentequilibrium equations in the limit as 6s+O.

a'( I + + k x V(s + ) + (s) - 0 (19)s

To obtain the scalar equations represented by Eq. (19), N(s+) is expressed interms of its vector components and the Serret-Frenet formulae are used to

transform the derivatives of the unit vectors

da.i + j + M --

- a + j + a k + M x yd zd

3M + M 3Ma * + j J + a-i k + M (xj - Kk) + M (-A) i

+ M 1. (20)z

Also, since

19

In Figure 4, Ra0 , R'o and P are position vectors of points, at positions" 8" "a + 6s" and "a + i" relative to a fixed point Q. If R'(s) is the

position vector of any point in the space curve relative to 0 we may write

It# at R' -0 0

R~ + d/ds 16. +1 d2R .(6s) 2 + - R

duds + 6/s *s 6s ~ 2 (ds0-2 2 ds 2 ..

k 6s + K/2 i (6s)2 (15a)

since by definition k = dR/4s and by Eq. (10a) dk/ds K C i.

In a similar way it is found that

R =P - R0

2kL + K/2 i C (15b)

Combining Eqs. (7) and (15b) yields

i x T - k(s) x T(s) C + [K/2 i x T + k x L I +

Therefore,

s+69s 6s 6s

fJ x T de - k x T f dC + [K/2 ixT + k x aT/as] f 250 s

" k(s) x T 1/2 (6)2 . (16)

Next,18

.. .. . . ". -- - .*.*. . .-.-..... .. -.. ..... .... ....-...... . . "..... . ..- '. .'..'.. - . . -% % '........ .. ? ¢ : ; ; . . . ; . _

. -__ 4 - • . ' -- . . .. .-

av av- + (.cV + - V,s X z as y

+ (XV + av /as)j + T i + T j + T k -0. (1I)

Finally, Eq. (11) may be written as three scalar equations which express the

condition of force equilibrium for a beam element

a V/ Ps - .V + KV + T - 0, (12a)Ky z x

a V/ Ps + XV + T - 0 ,and (12b)

aV /as - KV + T - 0 (12c)z x z

Next, the conditions for moment equilibrium are formulated from

consideration of the free-body diagram of Figure 4. Summing moments about

point 0 yields

M(s) + M(s + 6s) + K' xV(s + 6s)

s+6s s+6s

+ f xT(s)ds + f a ds 0 0 (13)

a 5

Again the terms appearing in Eq. (13) must be expressed in terms of theirvalues and those of their derivatives at "a"

*(s + - M(s) + s! + a.-. C2/2 + .. and

s+6s 68s 68 2 s 2

f ads = s(s) f d; + 2)"If d; +2 Is f d4 +8s 03 0as 2 0 'as2

- ()2 + ... (14)

1.7

f Td T(0)s + I 1. (6s)2/2 + *.. (8)

sa

Returning to Eq. (7), we next express V(s + 6s) in terms of its value

and those of its derivatives at 6s - 0 using vector Taylor's series.

V(s + 68) - V(s+ ) + + 6+ + ... (9)5

The symbol "s+11 is used in Eq. (9) as a reminder that this force acts on a

positive face of the element. Therefore,

Ws-+ V i + V j + V k and (9a)x y z

v I av V aV di dk

3s+ as as as X ds y de z ds

(9b)

Expressions for the rate of change of the unit tangent, principal normal

and binormal vectors with respect to arc length are given by the Serret -Frenet formulae (Appendix A) as

dk/ds = /p i - K i ,(10a)

d - Xi and (lOb)ds

di/ds = Xj - Kk (10c)

where K(- l/p) is the local curvature of the line of centroids. Upon

substitution of Eq. (8) through Eq. (10) into (6) and division by 6s, the

following vector equation results in the limit as 6s approaches zero,

16

LM(s+ 8s)

0,of W(s+ 8s)

T: R

' Q

'R5

°-.

,,Figure 4. Free Body Diagram of Ib Elmet

';: 15

The surface tractions and body forces may also contribute to a moment

resultant per unit length about the three coordinate axes. This moment

resultant is defined by

, - mxi + m j + m k. (5b)x y z

For problems involving motion of the rod element, the force T willcontain terms involving inertia forces while the moment a may contain moments

. of inertia forces about the three axes.

A free-body diagram of a rod element is shown in Figure 4. In Figure 4,

points 0 and 0' are again points in the curve of centroids at positions "s"and "s + 6s" respectively. The force and moment resultants acting on the rod

., cross sections of these positions are indicated as well as the surfacetraction vector and surface traction moment resultant at a typical point

- s + 4 where s < s + C s + 6 s.

R is the position,vector of the point x + relative to 0 while R isthe position vector of 0 relative to 0.

The condition of force equilibrium applied to the rod element yields the

- following equation

s+t sV(s) + V(s + 6s) + f T ds = 0. (6)

Now,

DT I; +_ 2jT 82+T(s + =T(s) + - (7)

a + ' a2 + ..2 7

as

where

s <(s + ) <(a + 6S) and ds d•

Therefore,

f T ds = T(s) f d-2 f s s 2 s s "+ "oraas

14s o0

..

Mzz

44

m

-

H - f(xz - y )dA (2c)4bz yz xz

Based on the above definitions, the vector force and moment resultant ofthe tractions acting on the face with centroid "0" at point "a" on thespace curve of centroids is

V(s) - - V (s)i(s) - V (s)j(s) - V (s)k(s) and (3a)x y S-

M(s) = - M(s)i(s) - M (s)j(s) - M (s)k(s) . (3b)x y z

Similarily, these vector quantities acting at a + 6s on the face with centroidS0' are

V(s + 6s) = V (s+6s)i(s + 6s) + V (s + 6s)j(s + 6s)|x y

+ V (s + 6s)k(s + 6s) and (4 a)z

N(s + 6s) = M (s + 6s)i(s + 6s) + M (s + 6s)j(s + 6s)x y

+ M (s + 6s)k(s + 6s). (4b)z

These quantities are illustrated in Figure 3.

Note that the unit vectors used to describe the directions of the vectorforces and moments at s are different from those at s + 6s since thedirections of the space curve unit vectors are constantly changing along the

curve.

The resultant per unit length of the tractions acting on the lateralsurface and of the body forces acting in the interior of the element is given

by

T = T i + T j + T k (5a)x y z

where Tx, Ty, Tz have the dimensions of force per unit length and i, j, k arelocal unit vectors.

12

',, ... ,..,:.'. '.,, ,,' .. '..',.-.. .'.' " . . .'. v.. • ... ... 4, % . .. . , .. ,. . .-.. , . ,- ~..-, - .,

Figure 6 illustrates the initial and final positions of these materialpoints as Q and Q' undergo displacements and un,. The magnitude of the

position vector of Q' with respect to Q, r is fould by use of the followingvector equation relating the initial and final relative position vectors ofthese points to their respective displacement vectors,

r + UQ, UQ (4)

, The displacement components u, v, and w at point Q are again given by

UQ W u i + v j + w k. (5)

The displacement vector at point Q', adjacent to point Q, can then be

expressed as

u, U. +- 6x + 6Y+ (6)QI Q ax ay az

where higher order terms in this Taylor's series expansion of the displacementvector about point Q have been dropped. The vector derivatives appearing in

Eq. (6) are next presented in terms of their vector components,

au "x 3L + LvJ + 1- k (7a)

ax axa+ x ax

au au + v j + 1w k (7b)

By ay 8 a+

au au . v aw ki-"aikL . T I + -1-v + .1w k u " + v + w

au . v . wT . + I- + k + u (X - k) - vX i + w K or" z az 3z u o 0 0 o0 ,o

26-2~

.- . .

au aak 7

L2 -a )v + K V) i + (- .!+. u) J + (2-w c~ (7c)az z 0 za 0

where the Serret-Frenet formulae have been used to express the rate of change

of the local coordinate unit vectors along the curve in terms of the localcurvature and torsion.

Substitution of Eqa. (5) through (7) into (4) and using Eqs. (3) yields

the following result for the position vector of Q' relative to Q:

[r(l +L2-l + m + n (2 - XV + Kow)] Ma az 0 0

+ v m(l + L) + n(Xu + L--)] j

aw aw+r[ - + m y + n(l-iKu + -)L k. (8)

ax" 0 az

If the magnitudes of r and r se denoted by r and r respectively, then theunit elongation, e, of the original line segment is defined by

r - r(l + e) . (9)

If e is a small quantity, then

-2 2S r + 2e) (lOa)

or

-2e "/2 (rO (b)

r

Upon squaring the magnitides of the vector r given by Eq. (8),

' substituting the result into Eq. (lOb), and grouping the resulting terms as

o coefficients of the products of the direction cosines X, m and n, the

*elongation, e, may be expressed in the following form,

27

e,

. - :n... . .. .",.;.:-.:-?.. '.;'. ., ;'. ..

e + t2 + 2 C n+2 + 2m+2 In + 2e am (11)xx yy zz xy xx yz

where

ax +u [u 2 av 2 aw 2 (12)xx ax X + V) x

£ - .Lv + "2 [ (2u 2 + (3 Vy2 + (4!y , (12b)

2 + ;( 2 ) + 2 + A 2 + (. 2XJ2 2 A3K3V]+12[~ +A ) A v +~ vc - 2 A v

+/21( 0 0 0 0 0

3u 3v 3w 3-2 AV - + 2 A u - - 2o u 2- + 2 o w , (12c

0 z *0 a0 z0 z 1c

2 2-u + v + u + !W aw (12d)xy ay ax ax ay ax ay ax ay

2 e = U Bu+ -W v + K W + 2u 2u + Lv avxz az ax 0 0 3x 3z ax az

_-+3w 3w 3u 3u 3v w+ 2x- z ov-x + o W-L-W + xO U ou- u (12e)

2 e 3v + 3w + u + 3u au + 2v_ +2W w

.z 0 ay az ay az ay az

(12f)

+ a v _w VLU + K W Uo y 0 ay 0 ay 0 ay

By definition, the quantities C £ C , £ , and e are the normaland shear strains at Q referred to t ye local coordinates at K Eqs. (12)therefore constitute the required strain-displacement equations.

28

• :,~~~~~~~~~~~~~~~.-.......:.,. _................;......................................-...'..'.-........................

Clearly, various approximate forms of Eq. (12) may be formulated

depending upon the simplifying assumptions appropriate for a given problem.

These formulations may postulate small deformations with large displacements;

small deformation with small displacements but large curvatures and/or

torsions; small curvatures and torsions; or various other combinations of

simplifying assumptions. The reader may verify that the technical theory of

curved beams, in which the normal strain is a non-linear function of the

transverse coordinate, requires retention of terms involving squares ofcurvatures and of displacements. It is also noted that consideration of

buckling or of stretch-stiffening effects requires retention of non-linear

terms in Eq. (12).

The simplest form of strain displacement equations which contains the

effect of initial curvature and torsion of the centroidal curve is obtained bydropping all non-linear terms in displacements and displacement gradients.

The strain-displacement equations then become

aue =x ax (13a)

xv£ av -- (13b)

yy ay'

E aw . c - (13c)zz az 0

£xy 1 1/2 (r + )x, (13d)

1 -/2(- + !w- v + Iow) ,and (13e)xz3z a00

1, /2 [ + w u) (13f)yz z ay 0

* At this point the functional form of the displacement components in terms of

the coordinates is completely unspecified. In the next section the strain-displacement equations are put in a form which corresponds to the usual

technical beam theory.

IV. A TECHNICAL THEORY OF M)DS ITH INITIAL CURVATURE AND TORSION

- A. Displacement Functions

A linearized "engineering" theory of curved rods is next developed on thebasis of assumptions employed in the strength of material formulations for

. the bending and torsion of prismatic bars.

29

% , *,,.. j...%. ., . ... .. , . .. ... ,. ., .. . . . ,. . ... . . . . . . . . . .- : . ,- . .- . . . . - . -.

The fundamental assumption of both technical beam theory and that of

torsion of straight rods is cross sections unchanged in size and shape after

the structure is bent or twisted. It is not necessary to initially postulate

that cross sections remain normal to the bent centerline.

A typical cross section is therefore assumed to undergo a small rotation

about each of the three coordinate axes while its centroid is displaced alongeach of these axes. These assumptions imply the following form for thedisplacement functions,

w(x,yz) - w 0(z) - x a (z) + y a (z) , (la)o yx

v(x,yz) v 0(z) + x *(z) , and (ib)

u(x,y,z) - u (z) - y (z) • (1c)

In Eq. (1), U0, vo and wo are the displacement components at a point on thecentroidal curve (the origin of the local x,y coordinates) whilea , a and * are small angles of rotation about the x, y and z axes in that

orer, as shown in Figure 7.

It is noted that theory of torsion of bars with non-circular crosssections requires the addition to Eq. (1) of a term representing warping of

the cross section. Introduction of such a term would greatly complicate theensuing development with very little effect on the state of stress other thannear a fixed boundary. For this reason, the displacement functions of Eq. (1)will be employed irrespective of the shape of the cross section of the bar.

Warping of compact cross sections increases as the ratio of the principal

second moments of area differs increasingly from unity. The warpinfphenomenon is thoroughly treated in texts by Timoshenko and Goodier and bySokolnikoffO

B. Strain-Displacement Equations

The linearized form of the strain-displacement equations derived inSection III are used as the basis for the strain-displacement equations forthe technical theory.

S. Ti oshenko and J.N. Goodier, Theory of Elasticity, 2nd Ed., McGraw-

Hill, New York, 1951.

6 I.S. Sokolnikoff, Atthematical Theory of Elasticity-, 2nd Ed., McGraw-Hill,

New York, 1956.

30

.i.. . ..y.2.... ........... .....4¢ ...- ¢.- . -. .. ¢-..- . ¢ . .¢.-.° - ' -' f : ' '7 ' ' ' ' '%. f %.%% .-'-ss-:- . , .- * *** J , *. ,*

ayay

Figure 7. Displacement and Rotation (baponents

31

* ~ J*~jw P P ~'* . ~ */

Substitution of Eq. (1) into Eq. (111-13) yields the followingexpressions for the components of strain,

e - en M 0, (2a)xx yy xy

aw aa 3azz . az X z + y z - (u - y , (2b)

2c z a - X v + K w + y(K - - x(x + ) , andXZ Z Y .0 0 0 0 X Z 0 (2c)

av2 X u + a + x -y X . (2d)ayz z o x z 0

Eq. (2a) represents the condition that cross sections do not deform andthereby confirm the satisfaction of the initial displacement assumptions.

Eqs. (2c) and (2d) show that transverse shear deformation cannot be madeto vanish for all values of x and y. Introduction of the angle * to representtorsional rotation would lead to this result even for a straight bar. For thecurved bar it is noted that the presence of initial torsion and curvatureprevent these strains from vanishing everywhere even with the omission of the

rotation *.

The condition that cross sections remain normal to the centroidal curve

can be invoked locally at x-0, y-0 or, alternatively, in an average sense byintegrating Eqs. (2c) and (2d) over the cross section. In either case, thefollowing equations expressing the vanishing of transverse shear deformation

(in the above meaning) result:

auo- 0 X +Kw and (3a)y az 00 00

av" u 3 (3b)

Another simplifying assumption often invoked is that the centroidal curve(x-y=O) remains unchanged in length during the motion of the rod. From Eq.(2b), this condition may be expressed as

awz0 ou 0o . (3c)

32

C. Inertia Force. and Nomeato

The inertia force resultants and couples per unit length are obtained byintroducing the assumed displacement functions into Eqs. (11-25) and (11-26)and carrying out the indicated integrations. It should be remembered that theorigin of the coordinate system is the area centroid of the cross section sothat integrals of the form f xdA and f ydA vanish . The resulting

A Aexpressions, assuming no variation of mass density within a cross section, arepresented below.

TIx M - u° A0 , (4a)

Tiy -YV A, (4b)

TIz - yw A, (4c)

Mix Y + 1 xya (5a)

ly Iy a yy a ,and (5b)

mi= - Y J p (5c)

I a f y2 dA , (6a)

I f xdA, (6b)YY A

I a f xy dA ,and (6c)xy A

J I + [ . (6d)p xx yy

33

.,. ....... ,..-.,-,.~~~~.-...-.,.-..-.-...-....-.-,.:...............-...:. .-...-..,...-................-....,....-....:... ?,.......

l and I are the second moments of area of the cross section for thecentroid a out the x and y axes respectively; Ixy is the product of area with

respect to the centroid for the x and y directions; and J is the polar moment

of area for the z direction with respect to the centroid.

The area and shape of the cross section of the rod may vary in an

arbitrary way along the rod. The orientation of the principal directions of

the cross section area with respect to the plan of curvature also may vary.

The quantities defined by Eq. (6) should therefore be considered to be

functions of the z coordinate.

D. Moment-Curvature and Force-Displacement Equations

The stress-strain relations which will be used for the derivation of the

equations relating the force and moment resultants acting as a cross section

of the rod on one hand, to the displacement and rotation components, on the

other, are predicated on the assumptions that the rod is constructed from a

linear elastic material which is elastically homogeneous in each cross section

and that the normal stress components in the directions transverse to the

centroidal curve are small in comparison with the normal stress component

a

The applicable stress-strain equations then become

a ffi E c , (7a)

o = 2G e , and (7b)

G a 2G . (7c)yz yz

Twe required equations are next obtained by combining Eqs. (2) and (3) with the

deFinitions of the various force and moment resultants given by Eq. (II-i),

M=f ya dAA A

aw D +dEf y - - IU - X Y + y + K dA

A 3z

_EI -y-z y + E I -xx - - + K o o) , 8a)

34

,'.','.'.'-'' 7.?.;'>..'.; "." ". .,, . :" , ., ., ,- ii,. .-- ,.",:.. ~ ,ci lli

M- f xO dA - E I a - E I x + K ), (8b)y yyaz xy 0

M =f (x a-y ) dA

(8c)- G J -- G I K a + GI K a

p az xxo0 x XYo0y

auVx=fA a xz dA - k AG& -5 a y - 0 V 0 + K W 0), (8d)

av

V - f a dA- k AG az + Xu+ ax) and (8e)

awV z dA - E A (- oUo ) .(8f)

A Zz0

The coefficients k and k appearing in Eqs. (8d) and (8e) are calledx. yTimoshenko shear coefficients. These are dimensionless quantities, dependentupon the shape of the cross section, which are introduced to compensate forthe fact that not only is the shear stress non-uniform over the cross sectionof the rod but its actual distribution is inconsistent with the function formsdescribing the shear strain-displacement equations. Values of the Timoshenkoshear coefficients are documented for various cross sections as, for example,by Cowper.

The complete system of equations constituting the linear, technical,theory of rods with space curvature is comprised of Eqs. (11-12) and 11-21)along with Eqs. (4), (5) and (8) of this section. For convenience of thefollowing exposition, Eqs. (11-12) and (11-21) combined with (4) and (5) arepresented below where, in a manner consistent with the employment oflinearized strain-displacement equations, the torsion and curvature are takenas those of the undeformed rod.

aV 82u

x5 X V y + V = yA o - T , (9a)a: o y o z at2 Ks

G.R. Cowper, "The Shear Coefficients in Timoshenko Beam Theory," ASME J.Applied Mechanics, Vol 33, Series E, No. 2, pp 335-340, 1966.

35

av a 2+ X V =yA 0 T (9b)

az 0 x at2 ys

- K V = A --P - T , (9c)

az o x at 2 zs

am a2 a 2a- zx - X M y + K °0 M z-V y - YI xx at2x - Yl xy at--y- m xe(9d)

ama 2a acY+XM + Ha-V -yI -- M an (e

az oy o y at 2 at 2 xs (9d)

aM 2u 2

H-'-+AM + V =- y yIy -_.Lm ,and (9e)az o X X t at 2

z K M - yJ - m • (9f)z o x pat 2 zs

In Eq. (9), the terms Txs, Tys, Tzs represent the portion of the surface andbody forces acting which are not acceleration dependent while mxs, mys and zs

are the corresponding parts of the couples of these forces.

To determine the motion of a rod with space curvature using thislinearized formulation, the system of twglve coupled equations consisting ofEqs. (8) and (9) above must be solved for the six force and moment resultantsand the six displacement and rotation components subject to appropriate

initial and boundary conditions. This formulation retains the effects oftransverse shear deformation, rotatory inertia, and extension of the

centroidal curve. Since some or all of these effects can safely be neglectedfor problems involving thin rods and low frequencies, more simplifiedformulations will next be considered.

It is noted that no restrictions have yet been imposed on the shape ofthe cross section, the orientation of the principal axes of the cross section

with respect to the local plane of curvature, or the change in thesequantities along the rod. The cross section area, the torsion and curvature,

and the tensile and shear moduli may also have arbitrary variation with z.

36

• . ..--. • . % .. . . .. - . .. . . * .. • . % -, - .. - - -. -. . • - , -. . • . - - , . - -, . % - %. .

X. Dieplacement Equations Of Equilibrium Reglecting Transverse ShearDeforaation And Rotatory Inertia

1. General Form

In this section the equilibrium equations for the rod are phrased in terms

of displacement functions. The effects of torsion, curvature and orientationof the principal axes of the cross section areas as well as the variation ofthese quantities along the rod are then examined.

In the following formulation transverse shear strains are neglected in

the sense defined by Eq. (3). Since rotatory inertia generally has lesseffect upon predictions of frequency and motion than has transverse sheardeformation, it is consistent to set terms involving a and a equal to zeroin Eq. (9) although rotational inertia effects about t~e z ax~s areretained. Equations (8d) and (8c) for Vx and Vy respectively becomeinconsistent because, although e and e are set equal to zero on thexz yaverage, equilibrium considerations require non-zero transverse shear

forces. These forces are eliminated from the equilibrium equations by solving

for V and Vx in Eqs. (9d) and (9e) and substituting these results into theremaining four equations. The equilibrium equations then become

M X - 2M + 2X M ' + X 'M + . K My 0 y o X o x OOz

(lOa)

-K V -yAu + T - m' +Xmo z o xs ye o xs

Mfg - 2X M' - ' M + (M)' yA v - T + m' + X mx o x o y 0 y o z 0 ys xs o ys

(lOb)

K M' + K X M + V; - YAw - T - K m and (10c)o y 0 0 x z 0 zs o ys

M' - KoMx 11 YJ m (10d)z ox p zs

In Eq. (10) superscript "prime" indicates differentiation with respect to z.Eq. (9f) is repeated as Eq. (10d).

37

To obtain the required displacement form of the equilibrium equations, thefunctions O and a are first eliminated from Eq. (8) by use of Eq. (3). Thexresulting expressigns for the moments and Vz in terms of the four displacementand rotation functions are then substituted into Eq. (10). The resulting

equations can be presented in the following form:

12 + U t A10 u + B1 2 v' + B11 V' 10

+ "' +C + D0 yJ11 o 10 o D12 1 D10 zs

(1la)

IV

24 0 23 0 22o A21o A20uo

IV + B it + B 0 + B +S24Vo 23Vo 22vo B21v 20vo

" C23Wo o + C22Wo' + C21Wo0 + C20Wo

+ D22 '' + D2 1 ' + D20 = -yAu + T - m' + m ,0 Xe y8 oaxe

(11b)

A34uIV + A 33u A32u o A 31u + A 30u

B IV + B3 + B + B3 vB34o + ,,33Vo 32Vo B31Vo B3Oo

C33w C32Wo C31o C30Wo

+ D32W D 31 + DAv - T - m' - X m and (l1c)

38

-,.°.- . . -.. .- -.. . ,- ........ . . . ..- ; - .'> ,i-' .'o -. . ..'.' .'.',?. --. - ,> "

A43o A42uo A41o A40uo

+ B4 3 v o + B4 2 v o + B41 v o ' + B40v o

+ + C41w

+ D4 1 ' + D4 0 f Aw - T + K m . (lid)

The fifty-eight coefficients appearing in Eq. (11) are presented inAppendix B. There it is seen that these coefficients depend upon thegeometric properties of the space curve described by the centroids of thecross sections, the geometry of the cross section and its relationship to thelocal vector principal lormal and binormal, the elastic moduli of thematerial, and the variat:ions of these quantities along the rod.

It is noted that iui the general case described by Eq. (11), all motions

are fully coupled since all variables appear in each equation. This means,for instance, that any exciting force will cause motion involving all fourdisplacement and rotation variables. Also, solution of the free vibrationproblem will yield mode shapes which will each have four corresponding natural

frequencies.

To aid in the understanding of how various geometric and stiffness

quantities interact to effect the rod motion, several more specialized cases

will next be examined.

2. Straight Rod With Variable Section Properties.

If X and k and their derivatives are set equal to zero, the following0 0

equations are obtained:

(GJp P J p - zs (12a)

(El u '')'' + (El v '')'' = - yAu + T - m' , (12b)yyo xyo 0 xs y

39

,n' ,'-°,o,' , " '.....................-..-........-..........-.... ...-..... ... .... .-

P1 X p" (A-5)

where superscript prime indicates differentiation of vector function P withrespect to z.Proof:

C, = P1 - P = h P' + L P'' +

2!

, = P2 - P = - h P ' +-T P '' - .

x = h3P'xP'' + ... , and

ICxC, = Ih3 1 Ip,'xPl + ...

Note that T and 0 are perpendicular to each other since P'.(P'xP'') = 0

**If z is the arc length s from a fixed point Po of C to point P, S is

then given by

d2

dP dP

ds x ds2 13 = .(A-6)

I d 2"ds

2

Proof: Since

dPI

dP d2P d dP2ds 2 1/ ds

Therefore P'(s) is perpendicular to P''(s) so that

IP'(s) x P''(s)l = I'1 IP"l IP"(<sl

**The plane passing through P and normal to 5 is called the plane of

curvature or oscillating plane of C at P.

Proof: If h is the length of arc between P and P; Then

l m LC - dh+O

The plane passing through P normal to C is called the normal plane of c at P.

2. Vector Binormal Of A Space Girve (0)

Referring to Figure A-2, P1 and P, are points on C either side of point Psuch that z increases by an amount h rom P to P1 and decreases by the sameamount from P to P2. C1 and C2 are position vectors of P1 and P2 with respectto P respectively

C

P C2

Figure A-2.

A vector binormal, B , of C at P is defined as

lira . (A-4)h O TCI x C

Agas[n, the sense of B depends upon the choice of scalar variable z used todescribe the position of P.

If P is the position vector of a point P on C relative to a fixed point0, the vector binormal can be expressed as

54

• .o,*jh~a .q~~o.*r~b %, .- m .- m =-m. l ,,lj.-o o' °,"Q' ,, , • ,o'o'%.'. % " , ,, - -limbs",

A vector tangent, T , to C at point P is defined by

T = lim(A-1)h.O

where h is the increase in z from point P to P. Note that the sense of T isnot unique since it depends on the choice of scalar variable z. It points inthe direction in which z increases along the curve.

**The vector tangent can be expressed as:

p1P = (A-2)

where P' is the derivative of P with respect to z

Proof:

C = P1 - P

where

P1 = P(z+h)

By Taylor's series for vector functions,

P(z+h) = P + P'h + P''h 2/2! +

so that

C = p' +1/h P + andS[P 2 + h P''P'' + ... 1/2

C

**If the scalar variable governing the position of P on C is the scalararc-length displacement of P relative to a fixed point P0 of C then

T = dP (A-3)d -

53

. . r ,. ..--ImJ,=., ,.,,,,,., ..= m .s,.,, ,;,.. . ,.. ,,.. '. ,,V.'s., ,.... .... .,. .".. . . . . . . . .'.. . . . . . . . . . . .. ..... .". ., ." .

APPENDIX A

SPACZ CURVE GEOMETRY

This appendix defines and presents relationships among various vector andscalar quantities used to describe a line curved in three dimensional space.These quantities include the vector tangent, vector principal normal, vectorbinormal, the radius of curvature, and the curvature and the torsion of acurve at a point.

Although the ensuing material may be found in many sources, thisparticular exposition is abstracted from the book by T.R. Kane.* Rigorousproofs, which have been omitted from this presentation in favor of intuitivearguments, can be found in that book.

1. Vector Tangents Of A Space Qirve (iT)

In Figure A-I, C is a curve in space, 0 is a point fixed in space while Pand P1 are points on C whose position along C depends on a variable z.Vectors P and P1, are position vectors of P and P1 respectively, with respectto 0 while C is the position vector of P1 with respect to P.

C

0 IP P

IF~ C

Figure A-1.

T.R. Kane, Analytical Elements of Mechanics: Vol. 2 Dynamics, AcademicPress, New York, 1981.

52

• o . . . . . . . . . . . . . . . . - -

APPENDIX A

SPACE CRVE GEOMETRY

51

3IlLIOGRAPY (continued)

2. Avaramudan, K.S. and Murthy, P.N., "Non-Linear Vibration of Beams WithTime Dependent Boundary Conditions," Int. J. of Non-Linear Mech., Vol. 8,pp 195-212, 1973.

3. Carrier, G.F., "On The Non-Linear Vibration Problem of the Elastic

String," Quart. Appl. Math., Vol. 3, No. 2, pp 157-165, 1945.

4. Crespo de Silva, M.R.M. and Glynn, C.C., "Non-Linear Flexural-Flexural-TorsionalDynamics of Inextensional Beams; I. Equations of Motion; II.Forced Motions,"J. Struct. Mech., Vol. 6, No. 4, pp 437-461, 1978.

5. Eringen, A.C., "On the Non-Linear Vibration of Elastic Bars," Quart. Appl.

Math., Vol. 9, pp 362-369, 1952.

6. Haight, E.C. and King, W.W., "Stability of Non-Linear Oscillations of anElastic Rod," J. Acoust. Soc. America, Vol. 52, pp 899-911, 1971.

7. Ho, C.H., Scott, R.A., and Eisley, J.G., "Non-Planar, Non-LinearOscillations of a Beam I. Forced Motions," Int. J. Non-Linear Mech., Vol.10, pp 113-127, 1976.

8. Ho, C.H., Scott, R.A., and Eisley, J.G., "Non-Planar, Non-LinearOscillations of a Beam II. Free Motions," J. Sound and Vibrations, Vol.

47, No. 3, pp 333-339, 1976.

9. Rodgers, L.C. and Warner, W.H., "Dynamic Stability of Out-of-Plane Motionof Curved Elastic Rods," SIAM J. Applied Math., Vol. 24, pp 36-43, 1973.

10. Shih, L.Y., "Three-Dimensional Non-Linear Vibration of a Traveling

String," Int. J. Non-Linear Mechanics, Vol. 6, pp 427-434.

11. Swope, R.D. and Ames, W.F., "Vibration of a Moving Threadline," J.

Franklin Institute, Vol. 275, pp 36-55, 1963.

12. Woodall, S.R., "On the Large Amplitude Oscillations of a Thin ElasticBeam," Int. J. Non-Linear Mechanics, Vol. 1, pp 217-238, 1966.

49

-. - ~ ~5- . - - . . %. .. .. .. - .. . . -. .. . . % - --- .- . . , .% - . .°

BIBLIOGRAPHY (continued)

13. Rutenberg, A., "Vibration Properties of Curved Thin-Walled Beams," ASCE J.Struct. Div., Vol. 105, pp 1445-1455, 1979.

14. Tabba, M.M. and Turkstra, C.J., "Free Vibrations of Curved Box Girders,"J. of Sound and Vibrations, Vol. 54, No. 4, pp 501-514, 1977.

15. Volterra, E. and Morell, J.D., "Lowest Natural Frequencies of ElasticHinged Arcs," J. of the Acoustical Soc. of America, Vol. 33, No. 12, pp1787-1790, 1961.

16. Volterra, E. and Morell, J.D., "Lowest Natural Frequency of Elastic Arcfor Vibrations Outside the Plane of Initial Curvature," ASME J. AppliedMech., Vol. 28, No. 4, pp 624-627, 1961.

17. Wang, J.T.S., "On The Theories of Ring Vibrations," ASME J. Applied Mech.,

Vol. 43, pp 503-504, 1976.

18. Wang, T.M., Nettleton, R.H., and Keita, B. "Natural Frequencies for Out-of-Plane Vibrations of Continuous Curved Beams," J. of Sound andVibrations, Vol. 68, No. 3, pp 427-436, 1980.

19. Wilson, J.F. and Garg, D.P., "Frequencies of Annular Plate and Curved BeamElementq," AIAA Journal, Vol. 16, No. 3, pp 270-272, 1978.

3. Non-Linear Strain-Displacement Equations for Beams

1. Maewal, A., "A Set of Strain-DisplacemenL Relations in Non-linear Theoriesof Rods and Shells," J. Structural Mechanics, Vol. 10, No. 4, pp 393-401,1983.

2. Kachanov, L.M., "A Brief Course in the Theory of Buckling," T. & A.M.Report No. 441, Univ. of Illinois, Urbana-Champaign, 1980.

3. Reissner, E., "On Otie-Dim.!nsional Large-Displacement Finite-Strain BeamTheory," Studies in Applie!d Math., Vol. 52, No. 2, pp 87-95, 1973.

4. Ericsen, J.L., Truesdell, C., "Exact Theory of Stress and Strain in Rodsand Shells," Archive for Rotational Mechanics and Analysis, Vol. 1, pp295-323, 1958.

C. Non-Linear Vibrations of Beams and Strings

1. Ames, W.F., Lee, S.Y., Zaiser, J.N., "Non-Linear Vibrations of a TravelingThreadline," Int. J. of Nan-Linear Mech., Vol. 3, pp 449-469, 1968.

48

I W7 7h- V 7- T 7,1 ~ - -

BIBLIOCRAPHY

A. leams and Carved eams-Linear Theory

1. Bishop, R.E.D. and Price, W.G., "Coupled Bending and Twisting of aTimoshenko Beam," J. Sound and Vibrations, Vol. 50, No. 4, pp 469-477,

1977.

2. Bishop, R.E.D. and Price, W.G., "The Vibrational Characteristics of a BeamWith An Axial Force," J. Sound and Vibrations, Vol. 59, No. 2, pp 237-244,

1978.

3. Chang, T.C. and Volterra, E., "Upper and Lower Bounds for Frequencies ofElastic Arcs," J. of the Acoustical Soc. of America, Vol. 46, No. 5 (Part

2), pp 1165-1174, 1969.

4. Davis, R., Henshell, R.D., ard Warburton, G.B., "Curved Beam FiniteElements for Coupled Bending and Torsional Vibrations," Earthquake

Engineering and Structural Dynamics, Vol. 1, No. 2, pp 165-175, 1972.

5. Den Hartog, J.P., "The Lowest Natural Frequency of Circular Arcs,"Philosophical Magazine, Vol. 5, Series 7, pp 400-408, 1928.

6. Dickinson, S.M., "Lateral Vibrations of Slightly Bent Slender Beams

Subject to Prescribed Axial End Displacement," J. of Sound and Vibrations,

Vol. 68, No. 4, pp 507-514, 1980.

7. Dym, C.L., "On The Vibrations of Circular Rings," J. of Sound andVibrations, Vol. 70, No. 4, pp 585-588, 1978.

8. Irie, T., Yamada, G., Takahashi, I., "Steady State In-Plane Response of a

Curved Timoshenko Beam With Internal Damping," Ing. Archiv, Vol. 49, No.1, pp 41-49, 1980. Also in J. of Sound and Vibrations, Vol. 71, No. 1, pp

145-156, 1980.

9. Irie, T., Yamada, G., Tanako, K., "Free Out-of-Plane Vibrations of Arcs,"

Trans ASME J. App. Mech., Vol. 49, No. 2, pp 439-441, 1982.

10. Markus, S. and Nanosi, T., "Vibrationl of Curved Beams," Shock and

Vibrations Digest, Vol. 13, No. 4, pp 3-14, 1981.

II. lao, S.S.,"Effects of Transverse Shear ind Rotatory Inertia as the CoupledTwist-Bending Vibrations of Circular Rings," J. of Sound and Vibrations,Vol. 16, No. 4, pp 551-566, 1971.

12. Rossettos, J.N. and Perl, E., "On The Damped Vibratory Response of CurvedViscoelastic Beams," J. of Sound and Vibrations, Vol. 58, No. 4, pp 535-

544, 1978.

47

..': . -.p. i . . a_ . . . .- X . -. , . ' , "z ° oI IZ.2, --

-, , , ,'. , .. -.."" ,. - i ~ ii .' ' , .

1. A.E.H. Love, A Treatise On The Mathematical Theory of Elasticity, 4th Ed.,Dover Publishions, New York, 1927.

2. L.M. Kachanov, "A Brief Course On The Theory Of Buckling," T&AM Report No.

441, University of Illinois, Urbana-Champaign, 1980.

3. E. Reissner, "On One-Dimensional Large-Displacement Finite-Strain BeamTheory," Studies In Applied Math, Vol 52, pp 87-95, 1973.

4. T.R. Kane, P.W. Likins, and D.A. Levinson, Spacecraft Dynamics, McGraw-

Hill, New York, 1983.

5. S. Timoshenko and J.N. Goodier, Theory of Elasticity, 2nd Ed., McGraw-

Hill, New York, 1951.

6. l.S. Sokolnikoff, Mathematical Theory of Elasticity, 2nd Ed., McGraw-Hill,

New York, 1956.

7. G.R. Cowper, "The Shear Coefficients in Timoshenko Beam Theory," ASME J.

Applied Mechanics, Vol 33, Series E, No. 2, pp 335-340, 1966.

45

V. CONCLUSIONS

This work was motivated by a requirement for a general analytical model of

a gun tube which could be used to examine the effects on tube motion of

centroidal axis curvature and twist, cross section eccentricity and variation

in cross section shape along the axis.

The model which has been formulated is based upon first principles of

mechanics and is sufficiently general to permit study of the above effects and

also of motions involving large displacements and rotations of the tube axis.

The model can provide useful information in its own right; both by

illustrating the manner in which the governing equations are coupled when

terms representing various affects are added or deleted, and by the free and

forced response solutions which may be obtained fur various special cases. It

can also be used to provide guidance for the development and use of more

approximate analytical techniques, such as finite element models, by

establishing the potential error associated with various approximations to the

actual structure. Since the derived equations show the coupling among the

displacement and rotation variables caused by axis curvature and cross section

non-uniformity effects, they can aid in the interpretation of physical

measurements of gun tube motion.

ACKNOWLEDGMENT

The author wishes to thank Mr. Alexander Stowell Elder for suggesting

that this problem was worthy of examination and for his patient endurance of

the sometimes-frustrated author's monologues.

44

•' . -. ".t.t. '. '.'.',. .' - " j .ig* ' . --".' , '.j ."

- A [EI + 2EI u ''' - Ao[2(EI )' + 2(EI )'I uo xx yy 0 xx yy 0

2 2 2-A [(El ), + A El + Ko2GIxx U 0 + K0oAo(GI x)' u - (EI v'')''

0 xx 0 XX 0 XX 0 00 XX 0

o (GI )' v '-2A0 K EI w '' - 2 K(EI )'w '

+ K0 (El + GJ ) ' + K [2 (El )'+(GJ )'] 'o xx p o) xx p

K(EI) 'A 2 K EI = + yA v- T + m' + A m , and (15c)o0 00 xx X oys

El u ''' + K (El ) u'' - 'I[A 2 + EA] u0 yy 0 0 yy 0 0 0 xx 0

-K (EA)' u - EJ V ''- K (EI )'v'0 o oa p 0 00 yy 0

+ (EA + K 2 El )w '' + [K 2(EI )'+(EA)']w o0

0 yy0o 0 yy~

2 "" (15d)o + K 0 E l x = A y w 0 - T z S + K 0 M .y!_o o xx 0 Zs 0 ys

Introduction of torsion again causes coupling among all of the equations

evern for a rod with symmetric cross section. It is also noted that although

the equations describing motion of a rod with constant cross sectional

properties would have somewhat fewer and simpler coefficients, they would

retain the coupling among the motions described by the four variables.

43

-.5

It is seen that while inclusion of effects of variable curvature somewhatcomplicates the coefficients and increases their number, the equation with Iequal to zero still describes two separate pairs of coupled motion involving

vo and * together and uo and wo .

5. Rod With Constant Curvature And Torsion And With Variable CrossSection

The final special case of Eq. (11) which will be considered is for a rod-* with constant torsion and curvature and variable but axisymmetric cross

section so that I remains zero. The displacement equilibrium equations forxy* this case are given below:

X K (G+E)I u ' + K0 X (GI )' U0 + K (G+E)I v " + (K OI )'v '0 X0 0 0 X 00 XX 0

+ GJ p " + (GJ)'' - Ko 2 El x x yJ p - m , (15a)

E" E IV + 2(EI )' u ''' + [(El )''-X 2 +21 )u - 2 (El)' u'

yy o yy 0 yy 0 yy xx 0 0 xx 0

+ [K + K Iu 0 - X (I + 21 )] v '

.

+ 2 X [(El )'+(EI )')V '+[X (El )''+X El +K X GI IV '+K El w '''o yy XX 0 yy 0 yy 0 X 0 0 yy 0

422K(EI )' w '' + [K (l )'' - K E1 - K EA~w0 yy 0 0 yy 00o yy 0o

+ [. 2El + GJ ]lb' + 2K ), (El )'$)oo xx p o o xx

u' =- A u + T -ian' + A m (15b)-0 xs y85 0 XS

, 42

: . _. A. .. . - ...

p~. 2K ( - K (EAu) K (EI v'')' + [(EA + KC EI)w ' 'o o x yy 0

K 0 2 (EI xy)' YAw - Tzs - Ko m ys (13d)

Eqs. (13) are seen to be fully coupled in the variables. If however, Ixy iszero as, for example, for a circular cross section, then two pair of equations

result; one pair involving vo and 4 only and the other uo and wo .

4. Rod Curved In A Plane With Variable Curvature But Constant SectionProperties.

The next special case is that of a rod for which X is zero but thecurvature completely variable. The principal normal and binormal directionsare assumed to be principal directions for the cross section. Eqs. (11)become

K 0o(G+E)IvxxVo0 + Ko0 GIXXv0' + Gp)' - 0 2 EI xx = yJ - m , (14a)

,1 IVu + K EAu + EI (w)'' - K EAw' yy o 0 0 yy 00 0 0

-Y~u + T -i' ,(14b)0 xs ys

E-." r v 0IV + GIX (KoV ')' +K 0 (El + GJ )4W'' +K 0 '(2EI +GJ )'.CXO xo o o xx p o xx p

+ K 0 EI 4 = yAv - T - m' , and (14 c)

SEI u ''' - EA (K u)' + (EA + K ZI )w 0 2 K 0 K 0 I w 00 yyO 00o 0 yy o 00o yy o

+ K K '' El w y YAw 0 T - K0 m . (14d)0 0 yy o 0 Es o ye

• 41

........................................... . . .

b - .

(El u ')'' + (El v 0 " yAv + T + m' , and (12c)-. xy o xxO 0 ys xs

(EAv')' - yAw° T • (12d)

Eqs. (12) are seen to be the usual equations of motion of shafts, beams andbars. Equations involving u0 and v are uncoupled if the x and y axes are the

principal axes of the cross section.

3. Rod Curved In A Plane With Constant Curvature And Variable SectionProperties.

In this case, the rod is initially in the form of an arc of a circle sothat K is constant and A is zero. The governing equations thus become

0 0

K0 El u '' + K (GI u ')' + K El v '' + K (GI v ')' +o xy o o xy o 0 xx o 0 XX o

2 2 2K El w ' + K (GI yw)' + (GJp*')' - K EI * yJ mO xy 0 XYo p o xx p zE

(13a)

2.2

(EIyy U ')'' + K0 EAu + (EIxy v ")'' + Ko(EIyy W ')'' - KoEAw ' - 0o(EI xy)''

-yAu + T -i , (13b)0 xs ys

2 2 2

(El u0K0 (GI u') +(EIyU "( v 0GI ' u + K v(EI( w))0xy o o xy o xx o o xx o o xy o

- (GI w )' - K (EI xx)o - Ko(GJp4b')' = - yAv + T - m'

(13c)

40

%,:.*~~

3. The Vector Principal Norml Of A Space Curve (v

The unit vector given by

V X T (A-7)

is called the vector principal normal of C oi P. Contrary to the casesof 0 and ir , the sense of V is unique.

An alternative expression for v in terms of derivatives of positionvector P with respect to coordinate z is

V=(P'xP'') x P' (A-8)I (P' x P'') x '

In terms of derivatives of P with respect to s. v is given by

- .(s (A-9)

This follows from the fact that P''(s) is perpendicular to P'(s) andP'(s) is a unit vector. The plane passing through P and normal to v is

called the rectifying plane of C at P.

4. The Vector Radius Of Curvature Of A Space Curve (p)

C

PFigure A-3.

56

Referring to Figure A-3, P is a point on C and P is the point of C withwhich P coincides when scalar variable z is increaseA by an amount h. C1 isthe position vector of P relative to P. By definition, the center ofcurvature of C at P is the point approached, as P1 approaches P by the centerQ, of a circle which is tangent to the tangent line of C at P and passesthrough Pie If r is the position vector of Q relative to P, the vector radiusof curvature, p , is defined by

P - Lim r . (A-10)h O

**In terms of the derivatives with respect to coordinate z of the positionvector, P, of P relative to a point 0 fixed in space, the vector radius ofcurvature is given by:

P M (P'xp'') x P' . (A-i)(p'xP')2

Proof: r(by construction) is perpendicular to both T and T x C1. Hence,*there exists some scalar, say 6 such that

r - 6T x (T xC 1 ) . (A)

On the other hand, as Q lies in the perpendicular bisector of. line P P1 I r may be regarded as the sum ofl/2 c1 and a vector perpendicular

to both c1 and rxc 1 . Hence, there exists some scalar, say U , such that

r - + uC X(Txc) . (B)

Dot multiply each of Eqs. (A) and (B) with cI (to eliminate 6), equate theresulting expressions for rcl, solve for 6 and substitute into Eq. (A) yields

C 1I 2 x 01 x C I

2 c • (* ) " ()

574.

.',

Next, use the relationships

X =-R7 ; c - h P1 + h2/2 ]P" +

and substitute Eq. C into Eq. A-10.

5. The Radius Of Curvature.

The vector radius of curvature p can be expressed as

p p v (A-12)

where p is an intrinsically positive quantity called the radius of curvatureof C at P which is given by

Sl .(A-13)

Proof: Multiply the numerator and denominator of the expressions for P of Eq.

(A-Il) with I ' giving

"W lp'l 3 (P'xP") x P'

By Eq. (A-8), Eq. (A-12) is an identity if p is defined by Eq. (A-13).

**In terms of derivatives of P with respect to s, p and p are respectively

"" given by

p Pff) and

I pf(S)I2(A-15)

1P

=

P 58

- - .a *

II ~ ~ ~ _% r I I ...;. -.. . . . . _ , . ... . . . .- , - - - - -- -.- = ,- ,

Proof: - and

[P'(s) x P"()] 2 - 1" >x P,,(8)1 2 " I?"<(s8)2

Furthermore,

(P'(s) x P"(s)) x P'(s) - (P) 2 P'' - '".?' P' P"(s)

6. The Serret-Frenet Formulae

The derivatives of the vectors T, 0 and v with respect to s are given by

dr vp, (A-16)

-- = X v and (A-17)ds

O d-/p (A-18)ds - ,an

'.5

where p is the radius of curvature and X, called the torsion of the curve c atpoint P is given by

x = p"[P'(s) • P''(s) x P'''(s)] . (A-19)

p.Eqs. (16), (17) and (18) are called the Serret-Frenet formulae.

'S Proofs:

59

-* * * **. - - - --.- - * * -

- P'(a)

T '( ) - P oo - p e - I "I v - v/pg i J(A-9)

8 - P'(s) x P''()[(P'')2 ]1 /2 , and

O'() = 'xP' " [(PI ,)21- 1/2,

_ p'xP,[(P') 2]- 3/2 e,,.p,,

Eliminate the first and second derivatives of P by using

P = T and Poo - v/p

Then

8' = p T x (p''' - v v-P'')

= p T x [v x (P'" xv)]

= p T- .P'.xV) V - p T V (P''xv)

Since,

Vo 0,

8' = py • (P'' x V) V

=-x ,

X= - p T • P'.xv)

p2 [p fxpt ip P'' ]''I •

60

.,..:.,. :... ._.:, ,.., ..... 4.:. .. . , . . .' . .. - , . *4. * ,, ,, * . . .. ,,...,:'.. , 4 ,

Finally,

V I i x T , and

V' =' x T + 0 x T

= - X V x + is x V/p

But,

V x T - (OxT) x T - B and

0xV jIx(SxT) = - K

Hence,

V' = A- r/p

**The torsion,A, expressed in terms of derivatives of P with respect to a

variable z other than s is given by:

A a P2 [P'(z) * P'(z) x PI"'(z)1 P'(z)1 - 6 (A-20)

Proof:

dz

P'(s) = P'(z) d

d2 2fdz 2 d! dz

P''(s) - P''(z) d + F(z) andds

2

61

dz 3 dz 2 3

Then,

P'(s) * P''(s)xP'" (s) = P'(z) . P '(z)xP'(z)(,-zs6

also

(dz)2 P,(z)2 = p,(5)2 1ds '( -- ( ) •

Therefore,

dz 6 [P(z)2 -3 -6

7. Example: The use of the Serret-Frenet formulas is illustrated by use of

the following example.

A straight line AC is drawn on a rectangular sheet of paper ABCD havingthe dimensions shown in Figure A-4. The paper is then folded to form a rightcylinder of radius a/w, the line AC thereby being transformed into thecircular helix H shown.

a. Determine the cosine of the angle * between a unit vector k parallelto the axis of the cylinder and the vector binormal of H at point P.

Solution: Let P' be a point on the cylinder base (Line AB) suchthat PP' is parallel to k. Let z be the distance between P and P', 0 theintersection of the axis of the cylinder with its base, n a unit vectorparallel to line OP', 0 the radian measure of angle AOP', and P the positionvector of P relative to 0.

62

zp

2b

Fl B

D

2 b

A PI

Figure A-4.

cos i k 8

*=P'(z) x p'"(z)

P a/it n + z k

63

P' =a/w k x n de + k

e (Since arc AP z ,andbb

dO 7F

dz b

Therefore,

P'(z) =bkxu + k

P"(z) = kx(kx-) d6

2-- u (Since kx(kxu) - n)

P'xP'' !a 2. k + n x k) andb2 b

2

IP' I " + a 1/2b b

Thus

2B k + n x k) (I + --- - 2

b b2

Finally,

C!.

Cos , - k*.- (I + b)- 1/2a

b. Determine The Radius Of Okrvature At P Of The Helix H.

P' xbkx n + k

2

P'xP - - (1 + a!12) 1/2 , andb b

a 3/2

- b 2 a (1 + b)

7r A- [+ a~J 2 ab 2b

c. Determine the torsion X of the curve H.

0 2[p'.p''xp''')[pt ]-6 ,

p' = b k x n + k

b2

,, , wa dal a2

'T"r - iT -3 lXl

b2 dz b3

65

-''" ', '"" .- '''"- '..- , ."" . . """"""". . . " -" -" -•• ' ' ' " ,-.'?-. ,- "-' -. '- '

r r~r.. - :- .

2

P'*(PV'xP"'') a 2

J

Then,

a 2+b2

8. Angular Velocity Of Space Ozrve O(ordinates In A Fixed Reference Frame.ds

Let P be a point moving along a space curve c with speed where is

the distance to P along c from some point of c. If R is a reference frame in

which the tangent (T), principal normal (v) and binormal (0) unit vectors to cat P are fixed then the angular velocity of R in the reference frame R' in

which c is fixed is given by

R'm R - (X T + 1/ p .) ds • A-21)

Proof: Given two reference frames R and R' and any two non-parallel vectors a

and b fixed in R. The angular velocity of R in R' is defined by Kane as

da db

R' R - t Xd t (A-22)u dadt b

If a and b are taken as the vector principal normal, v , and the vectorbinormal, 0, respectively at P then Eq. (22) may be written as

dv dOd--sx ds de

R' R = i~ Ts . (A-23)Ra dv dt

ds6

66

Next, the Serret-Frenet formulas give

dvd- 4 - 1/p T and

d- - X Vds

Substitution of the above into Eq. (A-22) gives the desired result.

67

!' ' t " ' .,,t.i., .,, .. ,,l,, .,w --.., . ,-.6 ,w

.. . ' a , , Wm k - ,l e , ' ' l '

APPENDIX B

APPENDIX B. COEFFICIENTS APPEARING IN EQUATIONS (IV-11)

69

APPENDIX B

APPENDIX B. COEFFICIENTS APPEARING IN EQUATIONS (IV-i 1)

A = K (G+E)Ixy

A 1 Xo 0K(O+E)Ixx +(KoGI xy)'

A Koo0Xo(GI xx)'+( o 'X o+K0 o ') GIx x+ K oX o' Elxy

B = K 0o(G+E)I xx

B = (K GI xx)' - K X (G+E)Ixy

B = - K X (GI xy)' - K 0 GI -K 0 '(G+E)I

C1 1 = 0 2 (G+E)Ixy

ClO =K 2(GI xy)' +K K ,(2G+E) Ixy

D12 =GJp

D = (GJ p)'

D10 - K 0 E1

A2 4 = Elyy

A3=2(EI )' - X El

yy 0 xy

A22 = (El yy) + 2Xo' Elxy - o2 E(21 xx+ yy)

70

A - A0(EI X,)'' + 4 A 0 (EI, )' + (3),0 ''-A.0 ) EI ,y

+ K 02 XGIY 2X 0o2 (EIXX)' -5 X 0 0' EIx

A0- o2 EA -2 X 0A '(El XX - (2 AL 0 X 0 1+0 2 )I

+ K GIX + Xo '(El x)'' + 2 XA '(El )'

+ -A 0 0 'A 02)EI

B24 ' Ixy

B '=2(EI P' - 2XEI -AX El23 xy 0oxx 0 yy

B22 2X0oE yy P X0 EIyy 2X0oE xx )

X-I'EI + (EI )I'+X 2 El0 xx xy 0 xy

B X (El ' 4 X '(El ) + (A 3 _3X '')EI21 0 yy 0 yy 0 0 yy

+ K 02 x0GI x+ 2,X02 (EI)' + 5 A 'ElX

B20 x0 '(Iyy )-X0 " Iyy P+(L0 x0 Ix0.. yy

+2A AO '(El )' + (2AO A ''+A '2)EI x-K 0 2 X 2 G1

71

C23 K0 EIy)'

C2 2Kc (El )' + 3Kc01E1 2), K ElI

2 0 0 yy 0 0 Xy

C 21 K 0(EI )'' + 4K'(EI)'y + (3K 0 1''A 02K0)I

-2K 0 (El )' (4X 0c K + X 'K 0 )EI x-K 0 EA

C 20 K c0 (EI y)'' + 2K 0 t (EI, ) ' + (KIll'-KIX 0A 2)EI y

-2A K% '(El )' (2X K ' + X0 'K ')EI00 xy 0 0 0 xy

* 3+o 0 GIXy

D K- El22 0 xy

D21 2X0K0 EIxx oo K JP 2 0 (IXy

-2 Kc I El0 Xy

-)2 2 K 0A (EI )' + (A 'ic0 + 2X 0 K' )EI x

K 0 (EI x)'' - 2 K 0'(EI x)' + (XO2 K O-K 0')EI x

72

.A 3 -Xo EIXX" 2 El -2(EIxy)'A33 0xy

I

A32 X- ' -3 (I' El - 2-(EI A ' ElA 2 0(Exx 0 0x 2oEyy) 0 yy

(Elxy)" - A 2E1 + o 2

A (EI 4 X '(EIxx)' + (xo3 _3o'')EIXX31 0 oExx) 0 4x 0 ox

" K 2 X - 2 x 0 (EIxy)' - 5 X0AX' EI

+: Ko 0 GI xyP+2K0K0GIx"+ 2(01) + 2 ,oc' GI,

A - - o ' (EI x x)'' - 2Xo''(EIxx)' + (Ao'Ao 2 -Ao''') El

+K2 (G~ )' + K (2 K 'X + K A ')GI

00xx 0 0 0 0 0 XX

-2000 ,+ 02) E~

2 '(EIx)' - (2 X EI0. 0 o oo 0 o xy

B -El34 xx

B 33 2(EIxx X 0 EI xy

2 2 2EB -- (El + E E K G 2 GI32 xx 0 xx 0 xx 0 yy

+ 2 o' (EIy)0 x

B 3 -2 K K GI + K2(G )' + 5 A A 'El31I 0 0ox 0 ox 0o yy

V73

1%

+ OX ' X 3) Elxy + K 2), GIXY

+ 4A0 '(EIxy)' + Xo(EIxy)II

B30 ' (2 A0'2 + 2 XAOO")EI + 2 Xo2(EIyy)Y

- (A ''f - x 2A o')EIxy - (2 K 0 'Xo + Ko2Xo')GIxy

+ 2 X '0 (EI xy)' - K2(GIxy )' + '(EI xy)''

C = - K El

C32 - -2 K EI - K ' EI -2 K (EI xy)'

C31 - - (2X ' + K 0 ')EI - 2)Ko (EI yy)'

+ (Ao2K - 3K o0 ')EI +K 3 GI - 4K '(EIx )'

- Ko(EI y)''

C30 - - (2OK '' + A ' K ')EI - 2 A K ' (EI yy)'0 00yy o

+ (x2K'- ...''')EI + 4 K 2 'GI

- 2 ''(El xy)' + o3(GIxy)' - K' (EI xy)''

D32 K (EIxx +GJP)

74

V. .. .' • p . .-_ 4 < %,%r

D31 ' K' (2EIxx + GJ) + 2 KX0 EIxy

+ 2 K (EI xx)' + Ko (GJ p)'

D30 ' (K ''A 2Ko) EIxx + (2A K ' + Ko 0 o ') EIxy

+ 2Ko '(EI )' -+ Ko (EI xx)'' + 2Ko A (EI xy)'

43 o yy

A - K-(EIy)'

2A42 'o EI+yyoEx)

A - 0 A0 Elxx - K EA + 2 A 0 xy + A 0 (El )Y

A4 = - 00 ' EI - K 0 EA0+ K 0 X EI40 000 xx 0 00 x

- 0o(EA)' + K A ' (EI xy)'

B43 K 0 xy

B42 A EJp + Ko (EIxy)'

B -- 2 K A ' EI - X K (Elyy)' + A 2 EI

B40 - ''ic EI - A 'ico(Elyy)' + ic A A ' EI

40 -0 0 yy 0 0 yy 000 xy

C42 0EA + EIyy75

C41 - 2 K 0 K ' Elyy + Ko2 (EIyy)' + (EA)'

- 0 2X EI X

C4 0 K K '' EI + K K ' (EI yy)' -K K 'X EI0 0o yy 00 o 00 0 o xy

2D 41 - K EIx

D40 o2Xo EI xx - KXK' EIxy - KO2 (Ei xy)

76

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12 Administrator 5 CommanderDefense Technical Info Center USA AMCCOM, ARDCATTN: DTIC-DDA ATTN: SMCAR-LCU, E. BarrieresCameron Station SMCAR-LCU, R. DavittAlexandria, VA 22304-6145 SMCAR-LCU-M,

D. Robertson

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US Army Materiel Command ATTN: SMCAR-SCM

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Alexandria, VA 22333-0001 SMCAR-SCA, W. GadomskiSMCAR-SCA, E. MalatestaSMCAR-SCA-T,

5 Commander P. BenzkoferUSA AMCCOM, ARDC SMCAR-SCA-T, F. DahdouhATTN: SMCAR-TDC, Dover, NJ 07801

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3 CommanderCommander USA Harry DiamondUSA AMCCOM, ARDC LaboratoriesATTN: Product Assurance Dir. ATTN: DELHD-I-TR,

SMCA-QAR-RIB H.D. CurchakD. Imhof DELHD-I-TR, H. Davis

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Commander 2800 Powder Mill RoadUSA AMCCOM, ARDC Adelphi, MD 20783ATTN: SMCAR-TSE, L. GoldsmithDover, NJ 07801 1 Commander

USA Harry DiamondLaboratories

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and Developme rt LaboratoryAmes Research Center 1 HQDA/DAMA-ART-MMoffett Field, CA 94035 Washington, DC 20310

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