54

---- TECHNICAL REPORT

WVT-720.

TRANSVERSELY ISOTROPIC BEAMS UNDER INITIAL STRESS

-- ~~~~~APRIL 1972-- • : "' i

AP 26 -97

S• ~~AIIClS Nlo. 501Al1.84400 , :

• •DA Project Nio. 1T0611OIA91A

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TECHNICAL REPORT

____WVT-7204

___TRANSVERSELY ISOTROPIC BEAMS UNDER INITIAL STRESS

_______BY

_____ EUGENE J. BRUNELLE

APRIL 1972

AMCMS No. 50IA11.B4400

VD Project No. IT061iDIA91A

BENET WEAPONS LABORATORYWATERVLIET ARSENAL

Watervliet, New York

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- REPORT TITLE

TRANSVERSELY ISOTROPIC BEAMS UNDER INITIAL STRESS

4. DESCRIPTIVE NOTES (rrp* of report and itnclusiv dates)

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S. AU THORS) (First name, EmiddiO init s o,. me)

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13. ABSTRACT

A linearized theory of transversely isotropic beams, which accounts for initialnon-uniform states of stress, is derived by perturbing and averaging (thru the thick-ness) the non-linear equations of elasticity. This theory provides a rigorous foun-dation for previously derived linear equations as well as displaying some new featuresconcerned with the stability and vibration of beams acted upon by a variety of initialstresses. A comparison is made to a previous theory (suitab.y one-dimensionalized) byHerrmann'and Armenakas which was obtained by variational methods and, although therc.. 'ts are vaguely similar, surprising diffez.•nces are found which support a previouscriticx, £ Masur.'

DD I'o 1473 OaSOLK FOR ARMY U::.

IUnc] jsiieds.*Ctty c',is~LUop __

14. LINK A LINK B LINK CSKEY WOROS- __ ___1

ROLM WT ROLz ! ROLM WT

Nonlinear Elasticity

InInitial Stresses

Beams

• ~Vibration

S~ Elastic Stabilitye

-4

•21

w

-•

Security Ca ssiflcttion

SS,.

TRANSVERSELY ISOTROPIC BEAMS LUDER INITIAL STRESS

Cross-ReferenceABSTRACT Data

A linearized theory of transversely Nonlinear Elasticityisotropic beams, which accounts for initialnon-uniform states of stress, is derived by Initial Stressesperturbing and averaging (thru the thick-ness) the non-linear equations of elasticity. BeamsThis theory provides a rigorous foundationfor previously derived linear equations as Vibrationwell as displaying some new features con-cerned with the stability and vibration of Elastic Stabilitybeams acted upon by a variety of initialstresses. A comparison is made to a pre-vious theory (suitably one-dimensionalized)by Herrmann and Armenakas which was obtainedby variational methods and, although theresults are vaguely similar, surprisingdifferences are found which support aprevious criticism of Masur.

I---- c

TABLE OF CONTEMrS

Abstract 1

Introduction 3

The Non-Linear Equations of Elasticity 4

"The Perturbed Equations 5

The Beam Perturbation Equations 7

Stability Problems 9

Vibration Probles 13

Comparison with a Previous Theory 19

Conclusions 20

Nomenclature 21

References 23

DD Form 1473 (Document Control Data - R&D)

1,Figures

1. Equilibrium of a Deformed Element 24

2. Equilibrium With the Surface Traction 25

3. Non-dimensional 3uckling Stress a Versus E'eG 26

4. Non-dimensional Frequency Rn Versus a 27

S. Non-dimensional Frequency n Versus a 28

2

IA

INTRODUCTION

During the past two decades significant idvances have been made in

the theory of non-linear elasticity. One of the potentially most useful

areas of research in non-linear elasticity, from an engineering viewpoint,

has been the development of smdll deformation equations superposed on a

state of finite deformation [l]. A logical extension of these results

would be to adapt these equations, by means of an averaging procedure,

to describe beam and plate behavior. For examiple the. spirit of this

suggestion has been carried out for plates by Herrmann and Armenakas [2],

who employed a variational method to derive their equations. An alternate

procedure, which is used in the present paper, is to start with the al-

,eady developed equations of non-linear elasticity and then to sequentially

perturb the equations, integrate thru the thickness, and finally introduce

displacement simplifications to obtain a linearized beam theory subjected

to an initial non-uniform state of stress. The advantages of this proce-

dure are two-fold. Firstly it permits some physical understanding to be

attached to the development of the governing equations and secondly it

provides a check on the equations (one-dimensionalized) developed by the

Drevious var-iational technique [2] and its associated assumptions. As

will be seen the two sets of governing equations are at wide variance,

and since the present technique is easily followed and checked one tends

to believe the present results. The details of this procedure are de-

veloped in the following sections.

t Obviously, this same technique may be used to generate a plate theory,which will be presented at a later date.

3

{

THE NON-L.IEAR EQUATIONS OF ELASTICITY

Referri-9 to -ijure 1 and following accepted procedures [3], the

stress vector equation of equilibrium of the deformed body is given by

a Ci l x i + = 0 (1)

V -* ". a t-s stress vector referred to the undeformed ith face area

aný is ,ihe t. 4. t'rce w"..-ctor referred to thc- undefo.med volume. Re-

sol•,"g ai* i'" Aon-orthogona, lattice vector (G )-directions yields

ai* = a. "* G. (2).1j 3

where a..* are the Trefftz components of stress, referred to the unde-1j

formed ith face area, which can be shown to be sytanetric [3]. Since

3RG - ax where R is the final 5. ate position vector it is seen thati4. aus

Gj=(is (6 a-x )is (3)

4. 44.where i s is the unit vector in the s orthogonal cartesian direction.

Decomposing X* into its 1 componentsS

4.-4.X* = i (4)

and putting the results of (2), (3), and (4) into (1) yields the follow-

ing scalar equations of equilibrium.

aus

+ , + Xs* 0 (5)ax i x. ij s

454

SNote that R r + u=(X + u ) is therefore .. , (6. + _.. s-. isX 35 axi x.i

Referring to Figure 2 where p* is the prescribed traction referred to the.?'. -). -

undeformed oblique face, and noting that dA (ii • n) = dAi/2, the equili-

brium of the deformed tetrahedron is given by

P*= n) 1 n 16 (6)

-' -4.

and by using (2) and (3) and defining the i components of p* as psS '

(6) becomes

P*= CF--* ni (6Js .- - (7)

Equations (5) and (7) are the desired non-linear equations that describe

the equilibrium condition and the surface tractions respectively. If the

extensions and shears are small, thp final areas and volume are equal

respectively to the initial areaF and volume so that a-i* = ,X* = X '

and ps* = p5 where a Xs, and p5 are the actual stresses, body forces,

and surface tractions respectively. These approximations are adopted for

the work that follows, so that the equilibrium equations and the surface

traction equations become,aus £

. s + ( x+ ) cj ] + Xs 0 (8)axi -- X ij .

aus

P ij ni js +--) (9)

THE PERTURBED EQUATIONS

Following a technique described by Bolotin [4] the following

quantities are introduced

us u +u (10)5 5

S

UJ

a o..a.+i .. (11

Ps =Ps + A PS + PS (12)

X=X + AX us2 (3Y,. = s + s +S - p P T• (13) .

where, for example, us, us, and us represent the final displacement, the

initial displacement, and the perturbation displacement, respectively.

The terms A ps and A Xs represent the change in the initial tractions

and body forces, respectively, due to the perturbation displacement.

Since the final equilibrium equations and traction expressions have forms

similar to (8) and (9) one has

au [ (6is + ')US . + Xs =0 (14)

~ a~sP5 i j ni (6js + - (15)

and upon entering (10) thru (13) into (14) and (15), and using the results

(8) and (9) one has

a u a - au aUs 5 NsS(a S) + N • (6o• + - + ) + +X + A•X• p -o - o 0(16)

3*53 x ax. ax.

- W5 au au-U (6.

P••s ý-)Is i(7s ij ax. ij js ax -- j ] ni (17)

auLinearizing the perturbation quantities, one neglects the term ai x

ii ax~Jand assuming the initial displacement gradients to be small [4] one neglects

aus

the term a- . This results in the following set of relations.

a ( ij -5j) +-- 2 - (i) +Xs +A•Xs - 5- = (18)

a- uPs+( Ps=[Yi -+ ti 0 n (18)

THE BEAM1 PERTURBATION EQUATIONS

Assuming a two-dimensional problem (plane stress) in the x -X3

plane (18) and (19) become upon expansion

+ X l (C AXI + aa I + a(20)

(a- -, -L - - -lý -" 1

axl ax_ ax, 33a3 x X

+X, + A X, - p • (20)

au 1 f u IU)a3 G a(- ;)-6• %C 3 + (01 ýL(

p, + A p, =alli 'U1 +F a13 U1 + 0-11 (22)

- 3-ax, (23)P3 + -A D3 al ax, aX 1 + a1•3 LU- + o••(23)

Now multiply (20) - (23) by dx3 and inteqrate thru the initial thickness'•T;

then multiply (20) and (22) by x3 d X3 and again integrate thru the thick-

ness. Into these results insert the following displacement and stress

assumptions,

u1 (xI, x3, t) = u (xl, t) + x3 i (xl, t) (24)

u3 (Xl, x3 %, t) = w (x1 , t) (25)1 au = E (u' +x 3 6'))-_ ax�1 La 3N K-= (26

0l 3 K2 G (ý- + -ý- K G (ip + w') (27)

to obtain the following beam equations of motion and associated boundary

conditions.

t Note that boundary conditions are desired only on t-he x, faces, there-fore ni '1I, n2 = n 3 = 0.

tit that is, the limits of all integrals are from- h/2 to + h/2.

7------ 4

(u' Nx), + (Wp' Mx)' + (4 Qx)' + E h u" + fx= p h i (28)

(w' N )' + K2 G h (ip' +w") + q =p h (29)x

(u' Mx + 1P' M.)' + (Ip Q.) + E12 - u' - 1p, Q. -Nzz - K2 G h (, + w') +mxh 3

SP Tf (30)

Fi +A Fi u' Nx +4 ' 1 .x + 1 Qx + E h u' or u=Uo (31)

F3 + A F3 = wI Nx + K2 G h (4, + w') or w =Wo (32)

M1 + A M= u' Mx + p'M* + p Q* + E Ty' or4,- o (33)

where Uo, Wo, and 4,o have been used to denote prescribed displacements,

primes denote differentiation with re:pect to x j , dots denote differentia-

tion wi 'h respect to time, and where the following definitions have been

used.

Sf (XI +A X)dx3 + Ci31 (u' + U31 -3 (u' -

+ I C(÷) M C(33]) 'P + (+) _ •-(-)

q = f (X3 + A X3 )CIx 3 + [ C31 - 031 ] w' + 033 0T 33 3

h + ) +(u -M= f (X- + A X-X 3 dX3 + 0 ( [3

(y3(+) + 4, + U31 + C31 ]

F. f Pi~ dX3 (i = 1,3) NX = f al i dX3F. = f A Pidx3 (i = 1,3) Nz = f CF3 dx3

t Note that there is a choice of stress or displacement boundary conditions.

8

MI - f P' x 3 dX3 Q = f a 1 3 dx 3

A 01 = f A Pl X3 dx 3 Q. = f C 3 1 x 3 dx 3

ax f all x 3 dx 3 H '1=fa I x2 dx 3

Note that the notation 33+- (h/2) and 033 - (- h/2) etc. In

the following sections the above results are used to analyze some stability

and vibration problems.

STABILITY PROBLEMS

This section employs static methods (i.E6. the "Euler Method") to

determine the stability of the various beam configurations considered. This

is an adequate approach if the loadings are conservative. Upon careful

examination it can be shown that the loading in Case IIl is non-conserva-

tive, hence the confirmation of that result must be postponed until the

corresponding dynamic formulation of that problem is carried out in the

next section entitled "VIBRATION PROBLEMS".

Case I, Just all = aN Acting

Equations (28) thru (33) reducei-• to

(ar + E) u" = 0 (34) N

( aN + K2 G) W" + • 2 G ' = 0 (35) '

(oN + h " - K2 G (iP + w') = 0 (36)

u (y) = '(y) =W (y) = 0 ; y 0 ,£ (37) •

t N is a constant

tt In all following examples, the boundary conditions are chosen to yieldthe simplest solutions.

9U

Neglecting (34), which leads to trivial results, it is assumed that

w W sWsin ir x/ k

IF= cos Ix X/

whence a non-trivial solution for W and T demands that det laijI = 0

where

all = (S - 1) a1/Z2 1-

az1 =- it/ a22 = (a- K) (S/K) - 1

and- Nx

S = (E/G) (h/L)2 K = (9,/Khh) 2 K2 = 7'/12 a = - UN K/E -ThTEuler

Hence the following equation for a is obtained where the lowest root is

desiped

S a2 -(S K + K+ l) a + K = 0 (38)

When S=O (i.e. G ÷ o) the result reduces to ao = K/(K + 1). Since usually

K > 100 this result is seen to be c!hse to the result obtained from a

Bernoulli-Euler beam analysis (Yo = 1). For S > 0, the qon-dimensional

buckling stress a is shown in Figure 3 where it is compared with the

classical result obtained in Reference 5. The close agreement justifies

the use of the simpler analysis developed in Reference 5.

Case II, Just-i- all = 2 x3 aM/h

Equations (28) thru (33) reduce to

E h u" +x(39)

K2 G h ('' + w") =0 (40)

i The classical result is a (I + S)-1i• m M is a constant

: 10

-L,* ~ .i* - , c'• • , . . . .. . ... ,•::• • ,

Eh 3p + u 0 2 G h (iP + W') =0 (41)

j ~*(Y) + Eh u' cy) Mti ui I + ýi Y) W (Y) =0 ;yO, k (42)

where Mx 2 2aM/12. Assuming that,

jw = W sin Tr~ x/

a non-trivial solution for W4, 'P, and U demands that det Iai j = 3 where

an1= E h a!: =Eh 0(a13)14

a~l= 0a22= KG h(iTZ) a23 K G h

a31 = M KI/)2 a3 C G h (i/a 33 K2Gh+E 3 T/)2/1

Expanding det Ia.ii= 0 yields the following result,

a r3E (43)

I Thus a buckling stress is predicted, even though it occurs at a stress

which is larger than that permissible for linear elasticity. However, a

simple plastic yield stress may be obtained by substituting the tangent

modulus E t for E in (43).

Case III, Just a13 = a Q Acting

Equtios (8) hru (33') reduce to

2 aQ11,' + E u" = 0 (44)

2~ G h ( + +w") 0 (45)

a Q is a constant

.... ... ...-...

E 2h2 - K2 G (, + W) 0 (46)

u (y) =' (y) =w (y) =0 ;y = 0,9 (47)

Assuming that,

u = U sin iT x/ Z

1P = T cos .n x/

w = W sin w x/ Z

a non-trivial solution for W, Y, and U demands that 0 where

all = E(ir/k) a12 = 0 a13 = 2 aQ

a 2 l = 0 a 2 2 = I/T A a 2 3 =

a)l = 0 a32 = K 2 G (7r/k) a33 = K 2 G + E h2• (7/Z) 2i12

Expanding the det JaijI yields

det a iji = E2 h 2 K2 (w/Z)6 G/12 t 0 (48)13

hence (48) demonstrates that a buckling stress does not exist.± The next

case demonstrates, however, that if aQ is applied conservatively, a buck-

ling stress does exist.

Case IV, Just, ;3 = a 0 Acting Conservatively

In this case fx : mx = 0 so that (28) thru (33) reduce to

*'' + E u"= 0 (49)

t The confirmation of this result is found in Case IV of the next section

entitled "VIBRATION PROBLEMS".

12

K2 G h (' +w") =0 (50)

h 21E - Q - K2 G (, + w') = 0 (51)

u (y)= ' (y) =w (y) -0 ; y=0,9 (52)

Assuming that

u = U sin Tr x/ Z

, = T cos -a x/

w = W sin 7 x/

a noti-trivial solution for W, T, and U demands that det Iai.! 0 where13

all = K (iIT/) a1 2 = 0 a13 = K aQI/E

a21 = 0 a2? : (7/r')IS a23 = 1/S

a31 = K a (,r/Y.)/E a32 = (Ir/Z)/S a33 = 1 + 1/SQ

Expanding det laijI = 0 yields the following result,

a EK(h/Z) (53)

This result has been determined previously by Herrmann and Armenakas [2].

However, as demonstrated by Case III, their statement "... we could assume

that the resulting shear forces either rotate with the element or do not

change direction after deformation." can not be correct.

VIBRATION PROBLEIMS

Case I, Longitudinal Vibration with Just all = aN Acting

In this case (28) and (31) are uncoupled and become

13

(aN + E) u" = pu (54)

u (y,t) =0 ; y =0, (55)

Assuming u = Un (sin n 7r x/ Z) e nt and letting aN = a c E the solution

is given by

n ( 1 +)12 (56)

'-n) ci

where

(W) = (E/p)n 1 2 nr/ k7•n cl

and

n = 1, 2, 3,

For some materials (glass fibers, for example) a may be as larbe as .1,

hence for this value of a it is seen that (56) may be in the range from

.95 to 1.05 for all values of n.

Casp II, Transverse Vibration with Just all : UN Acting

In this case just (29), (30) and (32), (33) are coupled and become

(UN + K2 G) w" + K2 G ' p w (57)1

(aN + E) h1 ip" K G (ip + w') = p (58)

1P' (y,t) = w (y,t) = 0 ; y=: O, (59)

Assuming that,

w = Wn (sin n T x/ e) nt

1P = T (cos n T x/ ewnt11 14

a non-trivial solution for Wn and T n demands that det Iai. - = 0 where

!2 a11 = n2 (S a - I) + f2 S az2 = - n £/r

a2 l = - n K i./. a 2 2 = n2 S (a - K) - K + S S2n

and

a2 p K (/IT 12 W2n/E

The frequency spectrum thus is given by

n [ (B [B - 4 A C]/ 2 )/2 A]/ 2 (60)

where

A=S

B = n2 aS + n 2 S (ay- K) - K- n 2

C n 2 a [ n 2 S (a - K) - K] - n4 (a- K)

When S ÷ 0 (zero transverse shear) (60) specializes to

= F (- n2 a K + n' [K - a] )/[K+ n2] ]I/2 (61)

Note that when a = S = 0 and K is large, S2n n2 which is the classical

result. Figure 4 plots the first two non-dimensional frequencies versus

a for three values of EIG with k/h = 10. The classical values [5] are

shown by circles and are in very good agreement with the present results,

thus completely confirming the results obtained in Reference 5.

Case III, Vibration with Just all = 2 x3 aM/h Acting

K- Equations (28) thru (33) reduce to

Vt Note that E/G = 0 yields the classical values and that E/G 2.6 yields

the isotropic values.

is

Mx + E h u" = p h u (62)

K2 G h (ip' +w")= p h (63)

El3 h3M u"+E " - K2 G h (P + w') = p -2 (64)

Mx (y,t) + E h u' (y,t) = r.1 u' (y,t) + IV (y,t)

= w (y,t) =0 ; y =O,. (65)

Assuming that

w=W (sin n r x/ . eiWntn

IP= T n (cos n 7r x,, 2) e n

ici tAu = U (cos n r. x/ £) e n

a non-trivial solution for Wn, n'i and Un demands that det laija = 0 where

c= K cUM/E and

all = K n - n a12- 0 ai3 = 2 n/K h .r2

n

a = 0 a2 2 = n -2 n S a 2 3 = n X/ln

a3l =2 n c/h a32 nTr K/Z. a3 3 =K (1 + n2 nS~

Expansion of the determinant yields the following expression for S1n"

K 2-_ f2) [ K- 2 ++(S12S^)2(K n- n' K - Q (n + K + n2 K S) - S Qn S- n2 ) (n2a )3 =0 (66)n n n n

For zero transverse shear S - 0 and (66) reduces to

(n2 + K) Q4 n -K n 2 (2 n2 + K) 12n + n6 (K2 a-2 /3)= 0 (67)

Noting that a appears only in the zeroth power term in (66) and (67) and

noting that uYM < .1 E (hence o < .1 K) it is seen that a has a negligible

effect on the frequencies.

16

1I NN - ,~ -. - ~

Case IV, Vibration with Just a1 3 a aQ Acting

Equations (28) thru (33) reduce to

Q u" = p (68)

K 2 G (q'p + w") = p (69)K2 G (Op + w') = p (70)

u (y,t) = P' (yt) = w (y,t) = 0 ; y = 0,Z. (71)

Assuming that

iW tuUn (sin n 7 x/ ) e nt

iW tI = r (cos n Tr x/ e) eint

w =Wn (sin n r x/1.)e i nt

a non-trivial solution for Un, Tn" and W demands that det Iaij = 0 where

a*= K a /E and

ia1 1 = il2 - K n2 a12 = 0 a13 = - 2 . n /i/T

a2 I = 0 a2 2 = S 112n - n2 a23 = - n Z/w

a31 = 0 a32 = - n 7r K/Z a33 S 2 n- K (1 + n2 S)

Expansion of the determinant, given below, demonstrates that a* does not

affect any of the frequencies and in fact the longitudinal frequency is

uncoupled from the other tNo coupled frequencies.

(I2 K n 2 ) [ (S a n2 ) (S 2 -K[l +n 2 S]) -n 2 K]=0 (72)n n n

For zero transverse shear S - 0 and (72) reduces to

t In particular no value of a* can make the frequencies zero or complex,hence instability -,annot occur as was predicted in Case III of the Dre-vious section entitled "STABILITY PROBLEMS".

1 17

n - Kn 2 ) [ S n4 - (n 2 + K[l +n 2 S]) Q 2 n+n 4 K] =0 (73)n n

Case V,Vibration with Just C13 aQ Acting Conservatively

In this case fx = m = 0 so that (28) thru (33) reduce toxx

*Q *' + E u" = p u (74)

*2 G (V' + w") = p (75)

E•- 72 -Q u' - K G (,b+ w') p pT- (76)

u (yt) ' (yt) =w (yt) =0 ; y =0, (77)

Assuming that

u U (sin n w x/ )eiWntn

• ='•n (cos n 7 x/) eiwnt

w =W (sin n 7 x/ 9) eiwnt

a non-trivial solution for Un, IFn and Wn demands that det laiji = 0 where

a1 n = -K n2 a12 =0 a1 3 = a* n £/w

a = 0 a22:S2 - n a23 = - n Z/r

a31 = n r, S K a./9 a 32 =- n K w/Z a 33 = S Q2 - K (I + n2 S)n

Expansion of the determinant yields the following expression for QnA

2) S g4 2 Sn2 2 f22 4 •.n (S S22 - 2)

(•2n ~Kn 2 )E[S n K (1 +nS) nn 2n * n

:0 (78)

For zero transverse shear S ÷ 0 and (78) reduces to

t •18

Q n = -B +(B2 -4 A C)I1/2]/2 A ]112 (79)

where

A=K+n2

B : - [ n4 K + n2 K (K + n2 ) ]

C = n4 K [ n2 K- a- ]

Figure 5 plots the first two non-dimensional frequencies versus a, for

three values of E/G with £/h = 10.

COMPARISON WITH A PREVIOIS THEORY

Referring to Reference 2 and one-dimensionalizing the resultst yields

the followir.g equations to be compared with (28), (29), and (30).

(V Qx)' + E h u" + fx p hu (80)

(w' Nx)' + K2 G h (,p' + w") + q p h (81)

Eh Q 2 h 3I x - Kx G h (ip + w) + mx P p 2 (82).-

It is noticed that the fori,_ of (81) and (29) agree, but that the forms of

(80) and (28) and of (82) and (30) have marked differences. Additionally,

Reference 2 leaves the term q, •- and to be arbitrary whereas the

present paper demands specific forms for fx, q, and mx as seen by the first

three results following (33). Also in a previous criticism Masur [6] has

pointed out' that a term (u' N x)' should appear in (80) which is accounted

for in the corresponding equation (28) of the present worlk. Finally, since

t Notational changes have been made so that the comparison of equations (36a),(36c), and (36d) of Reference 2 with (28), (29), and (30) of the presentwork may be facilitated.

19

II

the present work is based on simple straight-forward origins and is so

easy to check it is difficult to imagine that it can be in error; hence

one is forced to the conclusion that the one--dimensionalized results of

Reference 2 are not correct.

CONCLUS IONS

The several conclusions that may be inferred from this investigation

are that (a) with just aN acting, the simpler work of Reference 5 is ade-

quate to describe the problem, (b) with just aQ acting, very small changes

in the direction of loading exert a large influence on determining whether

or not the system can be unstable, (c) with just ao acting negligible ef-

fects are found, (d) transverse isotropy still exerts a significant influ-

ence on stability and vibration regardless of the type of initial stresses

that are acting, and finally (e) the one-dimensionalized version of a pre-

vious work [2] by Herrmann and Armenakas does not accurately describe the

present problem, hence doubt is cast as to the validity of their paper in

general.

20

Ab-_

7-i

NOMENCLATURE

E longitudinal Young's modulus

F + A Fi ; i = 1,3 perturbation boundary forces

fx" q9 mperturbation loadings

G transverse shear modulus

h beam thickness

° K = (/,/Kh) 2 geometry parameter

j beam length

M1 + A M1 perturbation boundary moment

Nx ,Nz Qx9 Q*' Mx' M. initial 9.,neralized stress resultants

P* generalized traction vector

PS actual traction components

Pi + Api perturbation boundary stresses

S = (E/G) (h/ X) 2 transverse isotropy parameter

u, w perturbation displacements

uV. wo perturbation displacements at the boundary"4 .X* yeneralized body force vector

X s actual body force components

( ) final quantities

S(-) perturbation quanti 'ies

-2 = r2/12 Mindlin's shear correction factor

perturbation rotation

Vo perturbation rotation at the boundary

Sgeneralized stress vector

aI I, a 3 perturbation stresses

./- K aE non-dimensionalized shear stress

K= K aM/E non-dimensionalized outer fiber stress

21

: - aK/E non-dimensionalized normal stress

aN, aQ, C constant initial stresses

a. Trefftz components of stress

ai actual stresses

Q non-dimensi onal frequency

22

22

REFERENCES

1. Green, A. E. and Zerna, W., Theoretical Elasticity, 2nd Edition, Oxford

Press (1968), p. 113.

2. HerrT,,dnn, G. and Armenakas, A. E., "Vibrations and Stability of Plates

under Initial Stress," Transactions of the American Society of Civil

Engineers, Vol. 127, Part I (1962), p. 458.

3. Washizu, K., Variational Methods in Elasticity and Plasticity, Pergamon

Press (1968), p. 52.

4. Bolotin, V. V., Nonconservative Problens of the Theorv of Elastic

Stability, The Macmillan Compary (1963), p. 43.

S5. Brunelle, E. J., "The Statics and Dynamics of a Transversely Isotropic

Timoshenko Beam," Journal of Composite Materials, Vol. 4, (July 1970),

p. 404. _

6. Masur, E. F., Discussion of "Vibrations and Stability of Plates Under

Initial Stress," Transactions of the American Society of Civil Enqi-

neers, Vol. 127, Part I (1962), p. 487.

.2

•_ ~23

r~

x'

+

:11

,0i I .I It10 .1

* i

}i

CC M

24

- -e --. - ~-~ n

IAT*dxd/Id

P~dA

R T~dxldx3/2 -

FIGURE 2

EQUILIBRIUM WITHi THE SURFACE TRACT1Oa4

A'A

25

II

1.0

40

.9 20

.8

.7-

10

.5-

.4

*CLASSICAL VALUES (REF.5) -5

0 10 20 30 40 50

E3

FIGURE 3

NO.-DIMENSIONAL BUCKLING STRESS o VERSUS E/C

1 __ ___ _______ ___26

I ~n=2 ,E/G=2.6n=,/=

4 ......

3- *LASSCAL ALUE

.6 .8 1.0.

1L10O-DMENSIONAL FIUNYQVERSUS

:17

4.0

38 -sI

3.6

22 n=2,E/G=50/ Il=1

n:I,E/G-50

01 2 3 4 5

FIGURE 5

'H ON-DIIIENSIONAL FREQUENCY n VERSUS

28