Dynamic stability of generally orthotropic beams

Fibre Science and Technology 13 (1980)15%198

DYNAMIC STABILITY OF GENERALLY ORTHOTROPIC BEAMS

C. C. H U A N G

Department of Mechanical Engineering, University of Western Australia, Nedlands, Western Australia 6009 (Australia)

SUMMARY

In this paper, a method of computing the instability regions of generally orthotropic beams has been formulated and the effect offibre orientation on the dynamic stability characteristics of the beams has been studied. In the formulation, both rotating inertia and shear deformation have been considered," the constitutive equations developed by Lekhnitskii have been applied; and the finite element method has been used. The numerical results indicate that the regions of instability vary erratically with 0 and that there appears to be more shift of the regions of instability for 0 < 45 o andfor higher modes.

NOTATION

A b

C,

cross-sectional area of beam non-dimensionalised natural frequency (pA/E:zlx)lJZLZo9 mutual stiffness coefficient torsional rigidity

Zi,

Y o u n g ' s m o d u l i (i = 1, 3, 4; or i = x, y, z)

Go h

shear moduli (i = 1, 2, 3 , j = 1, 2, 3; or i = x, y, z,j = x, y, z; i ~-j) thickness of beam moment of inertia about x axis polar moment of inertia about z axis stiffness m a t r i x of the complete beam geometric stiffness matrices of the complete beam kinetic energy length of beam 187

Cm,

Ix Jz [K]

[/%], [/¢,,1 KE L

Fibre Science and Technology 0015-0568/80/0013-0187/$02.25 © Applied Science Publishers Ltd, England, 1980 Printed in Great Britain

188

l M [M] P p* (]2 r2 SCOS 091

S~j

( . C. HUAN(i

length of elemental beam bending moment mass matrix of the complete beam stationary part of axial compressive force static buckling load generalised coordinates of the complete system

J:/(Al 2) l~/(Al 2) periodically varying part of axial compressive force material elastic compliance coefficients (i = 1. . . . . 6, j = 1. . . . . 6)

S2

(E=I.,,)/(KGr=AI z)

T

twisting moment time strain energy width non-dimensional total deflection of the centroid of beam crosssection (=y/l) principal coordinate axes of beam

I

U W Y (x, y, z)

Cd'Cmt ~p* = P tiP* = S

q

1 - {(E=I.,,Ct)/C2mt}

0

angle between direction of reinforcing fibres and the beam's longitudinal axis shape factor

~¢

;.

c,/(E=~A

2o

eigenvalues

p q5 q~0

density angular displacement of beam cross-section normal mode of q~ bending slope of beam cross-section normal mode of ~k frequency of axial periodic compressive force

z/L

qJo oJ

1.

INTRODUCTION

Fibre reinforced material is an important product of the continuing search for light weight materials with great strength and stiffness. The potential of fibre reinforced material has stimulated a considerable amount of study of mechanics of such materials in recent years. Generally orthotropic material, which is a special class of

DYNAMIC STABILITY OF GENERALLY ORTHOTROPIC BEAMS

189

fibre reinforced material, consists of much stiffer continuous unidirectional fibres in an arbitrary direction embedded in a relatively soft matrix. The problem of the determination of the in vacuo dynamic characteristics of orthotropic beams has been studied extensively by many research workers. However, relatively few papers are devoted to the study of dynamic characteristics of generally orthotropic beams. Using constitutive equations developed by Lekhnitskii8 and including the effects of shear deformation and rotatory inertia, free vibration problems of generally orthotropic beams have been formulated and studied by using Myklestud's method, 9'11 the continuous model method, 7'1° and the finite element method. 6'7 However, the dynamic stability of such a material has yet to be clarified. On the development of the theory of dynamic stability of bars, the problem was first investigated by Baliaev. 1 Later, Mettler 2 in his study introduced an additional term, the inertia force. The same problem was studied further by Bolotin. 3 The dynamic stability of beams with various boundary conditions was studied by means of the finite element method by Brown et al. 5 (using the classical beam theory) and later by Thomas and Abbas 4 (using the Timoshenko beam theory). The above work on dynamic stability has been limited to isotropic material. The object of the present research is to formulate a method of computing the instability regions of generally orthotropic beams and to study the effect of fibre orientation on the instability characteristics of the beams. In the formulation, both rotary inertia and shear deformation have been included; the constitutive equations developed by Lekhnitskii have been applied, and the finite element method has been used. The numerical results indicate that the regions of instability vary erratically with 0 and that there appears to be more shifts of the regions of instability for 0 < 45 ° and for higher modes.

2.

THEORY OF DYNAMIC STABILITY

Consider a generally orthotropic beam of any boundary conditions subjected to the action of the axial compressive force P + Scostot. To discretise the system, we divide the beam into a certain number of elements. A typical element is shown in Fig. 2. The problem being one dimensional, the orientation of the local and global coordinates is the same and the governing matrix equation of motion of the discretised system can be written as [M]{~/} + [Kl{q} - (ctP*[K~] + f l P * c o s c o t [ K g t l ) { q } = 0

(1)

where {q} = generalised coordinates of the whole system, [M] = mass matrix, [K] = stiffness matrix, [Kp], [Kst] = geometric stiffness matrices.

190

c . c . HUANG

[Kgs] and [Kgt] will be the same if the static and dynamic components of the axial load are applied in the same manner. The system equations of motion are obtained by assembling the elemental equations of motion and applying the boundary conditions. The elemental equations of motion can be generated by using a finite element approach similar to that used in reference 6. According to the theory of linear differential equations with periodic coefficients, 3 the regions of unlimited increasing solutions are separated from the regions of stability by the periodic solutions with periods 2~/t~ and 4~/co. More exactly, two solutions of identical periods bound the region of instability, whereas two solutions of different periods bound the region of stability. Furthermore, the regions of instability bounded by the solutions 47z/~o are the principal regions of instability which are the most dangerous and have therefore the greatest practical importance. As a first approximation, the boundary of the principal regions of dynamic instability can be determined from the following equation: (.o 2

([K] - (o~ _+ ½fl)P*[Kg] - ~ - [ M ] ) { q } = 0

(2)

if the static and dynamic components of the axial load are applied in the same manner.

3.

CONSTITUTIVE RELATIONS

According to Lekhnitskii, 8 a coupling effect exists between the bending and twisting moments. An element model of length lwith generalised coordinates at each node V (total deflection) ~b(bending slope) and ~b(angular displacement) is shown in Fig. 2. Referring to Fig. 1 and Fig. 2, the moment-strain relations can be expressed as m = (E=Ix/q)(O~b/dZ) + (CtEzzlx/qCmt)(t3¢/~Z) T = (Ct/q)(Odp/aZ) + (CtEzzlx/~lCmt)(~l/~Z)

(3)

y,z

3 z N Fig. 1.

A generally orthotropic beam under the joint action of a bending moment and a twisting

moment.

191

DYNAMIC STABILITYOF GENERALLYORTHOTROP1C BEAMS

I

I'

L

I

,t

Vi+t

V. !

Fig. 2. An element model. The torsional rigidity Ct can be approximated by zt Ct =

{1 + Fh/w}/{S'66 + F(S36/S33)(h/w)}

F = -0.6274{S'55/S;6} ~/2

(2n + 1 ) - S t a n h

~

T

(4)

($66/$5') /

n=0

(5) $33, $36, Sss, and $23 above are compliances of the fibre reinforced material referenced to the principal axes of the beam. They can be expressed in terms of the known basic material constants referenced to the axes of orthotropy by utilising the coordinate transformations of stresses and strains between these two sets of axes. The expressions of these compliances are given as follows: $33 = sin40/EI 1 + cos40/E33 + sin 2 0 cos 2 0(1 / G 13 - 2 vE 11/E33)/E11 $36 = 2(sin 2 0/Ell - cos 20/E3a) sin 0cos 0 + (I/GI 3 + 2rE11/E33)/Et t x (cos 2 0 - sin 2 0) sin 0cos 0 $55 = COS2 0/G23 + sin 2 0/G12

(6)

$66 = 4(1/E33 + 1/E11 + 2v/Ea3 - 1/G13 ) sin 20cos 2 0 + 1/Gla $23 = - ( v sin 20/E11 + v cos 2 0/E33)

4.

ELEMENTALMATRICES

4.1 Displacement functions Referring to Fig. 2, the displacements at any cross-section of the beam can be expressed as

192

c . c . HUANG {A(¢, t)} = [A(~)]{f(t)}

(7)

{A(~, t)} = {Y(¢, t)~b(~, t)q~(~, t)} r

(8)

{6(t)} = {Yi(t), ~bi(t), $i(t), Yi+ l(t), ~bi+ 1(0, ~b,+ l(t)} T

(9)

where

and the non-zero terms of [A(Q]i are a l l = (1 + 12s 2 - 12s2¢ - 3~ 2 + 2¢3)/(1 + 12s 2) al2 = {(1 + 6s2)~ - (2 + 6s2)~ 2 + ~3}/(1 + 12s 2) al4 = (12s2~ + 3¢ 2 _ 2¢2)/(1 + 12s 2) a15 = { - 6 s 2 ~ - (1 - 6s2)~ 2 + ~3}/(1 + 12s 2) a21 = ( - 6 ~ + 6~2)/(1 + 12s 2)

azz = {1 + 12s 2 - (4 + 12s2)~ + 3~2}/(1 + 12s 2) az4 = (6¢ - 6~2)/(1 + 12s 2)

(10)

a25 = { - ( 2 - 12s2)~ + 3~2}/(1 + 12s 2) a31 = {(6~/2)~ - (6~/2)~2}/(1 + 12s 2) a32 = {(3~/2)¢ - (3~/2)¢2}/(1 + 12s 2) a33= 1 -~ a34 = { - ( 6 ~ / 2 ) ¢ + (6a/2)~2}/(1 + 12s z) a35 = {(3~/2)~ - (3~/2)~2}/(1 + 12s 2) a36

=

4.2. Elemental elastic stiffness and geometric stiffness matrices The strain energy, U, of elemental length l, o f a generally orthotropic beam subject to an axial periodic force P is given by

U = i {½M(O¢/~z) + ½T(Ock/2z) + ½xAG(O¥/~z - ¢ ) 2 d z -

(~,Y/~z) 2 dz

(11)

U p o n substituting the constitutive relations (3) into (11) and non-dimensionalising, eqn (11) can be expressed in matrix form as

u=k(G/nl)fj {A'}r[D]{A'}d~-½Pl

f j {z~'}r{A'}d¢

(12)

DYNAMIC STABILITY OF GENERALLY ORTHOTROPIC BEAMS

193

where

{OY/~, ~'10~, ~¢10~, q, } [ .Is2~ 0 o --./S2~]

{a'} =

(13)

(14)

~/s22

0

0

rt/s22

0 0 0}

{ZX'}={~Y/~,

(15)

{A'} and {A'} can be expressed in terms of the generalised coordinates by differentiating eqn (7) with respect to 4; hence symbolically {A'} = [B]{6} and

{A'} = [B]{6}

(16)

where the coefficient matrix [B] and [B] are functions of~ and [B] consists of mainly zero elements. Upon substituting eqn (16) into eqn (11), one has

U = ½(C,/ql){6}r f~ [B]r[D][B]d¢{6} -- ½Pl{6} r f~ [B]r[B]d¢{6}

(17)

Differentiating U with respect to {3 }, the first term and the second term of eqn (I 7) will yield the elemental elastic and geometric stiffness matrices, respectively. 4.3 Elemental mass matrix The kinetic energy, KE, of an elemental length l, of a generally orthotropic beam of uniform cross sectional area is given by KE

fl {½pA~'2 + ½plx~2 + ½PJ~t~2}dz

(18)

Upon some algebraic manipulations, eqn (18) can be written as KE=½{~}T

pAl3[A]r[l][A]d~{~}

(19)

where the non-zero terms of [I] are 111 =

1

122

=

S 2

133 = 42

(20)

It is easily recognised that the integral in eqn (19) represents the elemental mass matrix.

194

c . c . HUANG 5.

NUMERICAL CALCULATIONS

The theory thus formulated is now coded in F O R T R A N language to calculate the region of dynamic instability and to study the effect of fibre orientation on the instability regions. The material parameters taken from reference 7 are

Ell~G12 = 3.7 E33/G23 = 30

GI2 = G23 =

0"3686 x 106 psi

0"6242

G13 -- 0 ' 7 4 7 9 x 106

v = 0'3

x 106

psi

psi

x = 0"8333

Other parameters are width = 1.2 in,

thickness = 0.125 in,

length = 7.5 in.

The evaluation of the stiffness and mass matrices follows the standard numerical procedure for the finite element method. Numerical integrations are performed to evaluate the expressions in eqns (17) and (19) and obtain the stiffness and mass matrices; then the elemental matrices are assembled and the boundary conditions are applied. A library eigenvalue subroutine from the West Australian Regional Computing Centre, EIGRF, is then used to calculate the buckling loads and the boundary frequencies. It should be mentioned here that in evaluating the static buckling load from the equation [K]{q} = P*[K~]{q},the computations break down because [Kg~] is not positive definite. The problem is overcome by modifying the equation to (1 + P*)[K]{q}= P* < [Kg~] + [K] > {q} by adding to both sides of the original equation a term P* [K] {q }; the matrix on the right side of the equation is thus made positive definite and the eigenvalue can then be computed. 6.

RESULTS AND DISCUSSION

The numerical results are obtained by using a 10 element idealisation of the clamped-free and simply supported beams. Figure 3 through Fig. 8 show the regions of dynamic instability for 0 = 30 °, 45 °, 60 ° and 75 ° It is seen from Fig. 3 through Fig. 12 that for both types of beam the first region is unchanged with varying 0; this is consistent with the results from reference 6, which indicate that the first region stays the same. In general there appears to be more shifts of the regions of instability for 0 < 45 o and for higher modes. For 0 > 45 °, shifts of regions do not occur for the hinged-hinged beam, as they do for the clamped-free beam. Only the first two regions are unchanged. The study of the values of P* and b with varying 0 also shows a similar pattern, i.e. there is drastic change for 0 < 45 o The interval of 0, where most shifts of instability regions occur, appears to coincide with the interval of high torsion-bending coupling. 8

DYNAMIC STABILITY OF GENERALLY ORTHOTROPIC BEAMS

195

.7~

.5

.25 0 ' 0 Fig. 3.

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i

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,

i

5

,

i

,

,

tO

.

i

L

I

,

f5

i

20

i

,

I

i

i

i

25

,

,

30

b/b,

Regions o f dynamic instability o f a generally orthotropic beam. (Clamped-free. 0 = 1 °, a = 0.5, P * ffi 2.45, b 1 ffi 1.432, r = 0.0722.)

.5

.25 0 Fig. 4.

,,,

~ . . . . . I0 . . . . . . . f5. . . . .

20

%0

b/bl

'

35

R e ~ o n s o f dynamic instability o f a generally orthotropic beam. (Clamped-free. 0 = ] 5 o, = 0'5, P* = 1, bt = 2.227, r = 0.0722.)

|

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~.5 .25

0 Fig. 5.

k

,'o

t'~ ....

~'o ....

3;'

Vb,

35

Regions o f dynamic instability o f a generally orthotropic beam. (Clamped-free. 0 = 30 °, a = 0"5, P* -- 0"41 I, b 1 = 1.432, r -- 0.0722.)

196

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,

,

,

,

,

i

,

i

,

l

,

,

,

,

,%,

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i

l

I

l^

75

5 25 .

l'

o

,I

o

Fig. 6.

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I

5

I

I

l

I

lo

Is

20

,

~o

I

i

,

b/b,

35

Regions of dynamic instability of a generally orthotropic beam. (Clamped-free. 0 = 45 °, = 0.5, P* = 0.252, b I = 1-123, r = 0.0722.)

1

,

I

i

,

,Av,

i

r

# .5 25

I

00

Fig. 7.

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,

i

i

i

,

lo

i

i

i

f5

i

i

i

i

20

' ~'0

b/t~,

35

Regions of dynamic instability o f a generally orthotropic beam. (Clamped-free. 0 = 60 °, ct = 0.5, P* = 0.199, b I = 0.999, r = 0.0722.)

.25[ °o

Fig. 8.

'

s

fo

'

' t~ . . . .

2'o '

'%'

so

~,/b,

as

Regions o f dynamic instability o f a generally o r t b o t r o p i c beam. (Clamped-free. 0 = 75 °, = 0.5, P * = 0-183, bl = 0.957, r = 0-0722.)

DYNAMIC STABILITY OF GENERALLY ORTHOTROPIC BEAMS

|

197

,

.75 I /I .5 .25 |

i

0 Fig. 9.

•

•

•

,

5 Regions o f dynamic instability o f a generally orthotropic beam. (Simply supported. 0 ffi 30 °, = = 0"5, P* = 1.04, b I = 3.988, r = 0.0722.)

I

.2 0o

5

Fig. 10.

IO

f~

*

i

20 J

I

i

,

*

25

'

I

Wh

I

30

Regions o f dynamic instability of a generally orthotropic beam. (Simply supported. 0 -- 45 °, = = 0.5, P* = 1.01, bl = 3"152, r = 0"0722.)

|

I

x

'

l

,

,

,

,

,

.75 1 .5

P

.25 " °o

Fig. 11.

. . . .

s

'

to

15

'

'z'o

. . . .

is

. . . .

b/b,

30

Regions of dynamic instability of a generally orthotropic beam. (Simply supported. 0 = 60 °, ,, = 0.5, P* = 0.796, bl = 2.803, r ffi 0.0722.)

198

HUANG

c.C.

7~ .5

I

.25 o

0

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I

l

l

l

,

5

,

l

l

J

lo

i

l

l

15

L

j

,

20

,

,

,

l

l

,

2S

,

l

,

_

b/b,

30

Fig. 12. Regions of dynamic instability of a generally orthotropic beam. (Simply supported. 0 = 75 °, = 0"5, P* = 0.730, b 1 = 2.684, r = 0.0722.)

REFERENCES 1. N. M. BAL1AEV,Stability of prismatic rods subject to variable longitudinal forces, Engineering Constructions and Structural Mechanics (1924) pp. 149-67. 2. E. METTLER,Biegeschwingungen eins Stabes Unter Pulsierendre Axiallast, Mitt. Forsch-Aust. GHH-Konzern, 8 (1940) pp. 1-12. 3. V.V. BOLO'nN, The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, Calif., 1964. 4. J. THOMASand B. A. H. AanAS, Dynamic stability of Timosbenko beams by finite element method, Journal of Engineering for Industry, 98 (1976) pp. 1145-51. 5. J.E. BROWN,J. HUTTand A. E. SALAMA,Finite element solution to dynamic stability of bars, A/AA Journal, 6 (1968) pp. 1423. 6. K. K. TEH and C. C. HUANG, The vibration of generally orthotropic beams, a finite element approach, Journal of Sound and Vibration (1979). (In press). 7. K. K. TF,n, The Vibrations of Generally Orthotropic Beams, Master of Engineering Science Thesis, Department of Mechanical Engineering, University of W.A., Nedlands, Western Australia, 1979 (to be submitted). 8. S.G. LEKHNITSKn,Theoryof Elasticity of an Anisotropic Body, Holden-Day, San Francisco, Calif., 1963. 9. R. B. ABAgCARand P. F. CUNNIFF,The vibration of cantilever beams of fibre reinforced material, Journal of Composite Materials, 6 (1972) pp. 504--17. 10. L.S. TEOHand C. C. HUANG,The vibration of beams of fibre reinforced material, Journal of Sound and Vibration, 51 (1977) pp. 467-73. 11. 1. G. RITCHIEand H. E. ROSiNGER,Torsional rigidity of rectangular section bars of orthotropic materials, Journal of Composite Materials, 9 (1975) pp. 187-90.