dynamic stiffness analysis of laminated beams using a first order shear deformation theory

0263-8223(95)00091-7

Composite Structures 31 (1995) 265-211 0 1995 Elsevier Science Limited

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Dynamic stiffness analysis of laminated beams using a first order shear deformation theory

Moshe Eisenberger, Faculty of Civil Engineering, Technion - Israel Institute of Technology, Technion City 32000, Israel

Haim Abramovich & Oleg Shulepov Faculty of Aerospace Engineering, Technion - Israel Institute of Technology Technion City 32000, Israel

In this paper the exact vibration frequencies of generally laminated beams are found using a new method, including the effect of rotary inertia and shear deformations. The effect of shear in laminated beams is more significant than in homogenous beams, due to the fact that the ratio of extensional stiffness to the transverse shear stiffness is high. The exact dynamic stiffness matrix is derived, and then any set of boundary conditions including elastic connections, and assembly of members, can be solved as in the classical direct stiffness method for framed structures. The natural frequencies of vibration of a structure are those values of frequency that cause the dynamic stiffness matrix to become singular, and one can find as many frequencies as needed from the same matrix. In the paper several examples are given, and compared with results from the literature.

INTRODUCTION

The increased use of laminated composite beams as movable elements of machines, such as rotating blades or robot arms requires the knowledge of their natural vibration charac- teristics. Most of the work done in this area was concentrated on calculations of natural frequencies of laminated plates’,’ and only a few dealt with beams. Vinson & Sierakowski3 calculated the natural frequencies and mode shapes for simply supported composite beams having mid- plane symmetry of the cross-section, based on the classical lamination theory which neglects shear deformation. For anisotropic beams the transverse shear deformation is very important because the ratio between the transverse shear modulus and the extensional one is about l/30. Therefore, the classical lamination theory fails to predict correctly the natural frequencies of laminated composite beams. Abramovich4 presented exact solutions for symmetrically laminated beams with ten different boundary conditions. This work is based on Timoshenko type equations, including shear deformation and

rotary inertia but with the joint action term of the two effects being omitted. Teoh & Huang5 also presented the influence of shear deformation and rotary inertia on the free vibrations of orthotropic cantilever beams, based on an energy approach. Chen & Yang6 introduced a finite element method to predict bending and free vibration frequencies of laminated beams including shear deformation. Exact solution to the problem of the free vibration of symmetrical laminated composite beams based on a Timo- shenko type theory is also presented7 for some arbitrary boundary conditions. Singh & Abdel- naser8 analyzed the equations of motion of a cross-ply symmetric laminated composite beam, using a third-order shear deformation theory. They showed that the results using a first-order (Timoshenko type equations) and third-order theory are almost the same. Recently, Abramo- vich & Livshits’ extended the approach of Abramovich4 to include non-symmetric layups. These are the only available results for non- symmetric laminated beams.

The exact element method” is used in this paper to solve generally laminated composite

265

266 M. Eisenbergel; H. Abramovich, 0. Shulepov

beams. This is possible since one derives the exact dynamic stiffness matrix, and then any set of boundary conditions including elastic connections, and assembly of members, can be solved as in the classical direct stiffness method for framed structures. When applying the exact element method, the two coupled differential equations of motion are solved, rather than the decoupled equations, as in Abramovich.4 Thus, there is no need to neglect the joint effect of rotary inertia and shear deformation, and thus the results in this work are used also to study the influence of this term on the vibration frequencies of a laminated beam, by comparison to the results in Abramovich.4

In this paper we extend the analysis for symmetrically laminated beams,’ to un-symmetrical beams. The advantages of the present method are in the ability to deal with general layouts and geometries of the structure and its boundary conditions with the ease of the general finite element method, using a minimal number of elements, but with exact results. Introduction of complex frame geometries and flexible connections is straightforward. From the com- putational aspect, the natural frequencies of vibration of a structure are those values of frequency that cause the dynamic stiffness matrix to become singular, and one can find as many frequencies as needed from the same matrix. In the paper several examples are given, and compared with results from the literature.

ANALYSIS

For harmonic vibrations, the equations of motion for laminated beams read”

d2u I,UPu+I~co2~+& -+B,, -=

dx2

d24 0

dx2

I,co3u +13CJJ24 +D,, - d2qb +B d2u

- dx2 ‘, dx2

+A55

(1)

(2)

(3)

where u(x) is the axial displacement along the beam, w(x) is the vertical displacement of the beam, dwldx is the slope of the beam (composed of two parts, C+(X) the bending slope and the additional shear deformation angle r(x)), and co is the frequency of harmonic vibration. I,, Z2 and I, are the mass inertia defined as

h/2

z,=c p dz (4) -h/2

“i h/2

I2=c zp dz -h/2

(5)

s h/2

I3i=C .z2p dz (6) -h/2

where p is the mass density, and c is the beam width. Also,

h/2

A,,=c Q,, dz (7) -h/2

s hi2

BII=C &,,zdz phi2

(8)

s h/2

D,,=c cj, ,z2 dz (9) -h/2

A55 =ck I

h/2

Q55dZ (10) -hi2

where k is the shear correction factor, and Q,, and Q55 are the transformed material constants as given in Vinson & Sierakowski.”

For symmetrically laminated beams, B,, = I, = 0 and the equations reduce to

d2u IIco2u+Ar, -=

dx2 0

I, co2w +A55 (J!$~)=o

(11)

(12)

(13)

Here we see that the first equation is un-coupled from the other two. These are coupled and are similar to the equations for the Timoshenko beam model that includes the effect of shear deformations and rotary inertia.4

If we normalize eqns (l-3) using the relation l=x/L, and choose for the solution the follow-

Stiffness analysis of laminated beams 267

ing polynomial series

cz ll=C Uiti

i=O cc

(14)

W=C Wi5i i=O

3c

i=O

(15)

(16)

-1 44-Z=

(i+l)(i+2)&

X (IlL2W2Ui+12L202fi

Wi+2=

(i +

f, 1+2

=_ (I~L202fi+I~L202Ui-~L2W2(I~Ui+~~f;:)+A55L(i+1)Wi+1-A55L2f;)

(i+l)(i+2)0,,

Substitution of these expressions and their derivatives in the differential equations yields

n; a; I*L2W2 C Ui<‘+12L2c02 Chti

i=O i=O

i=O

x:

+Bll C (i+l)(i+2)fi+2ti=0 i=O

(17)

i=O i=O

-A55L f (i+l)&+&=O i=O

(18)

i=O i=O

w

+Bjl 1 (i+l)(i+2) Ui+21i i=O

m

+a1 c (i+l)(i+2)fi+,iJ’ i=O

‘x

+A,,L c (i+l)wi+J i=O

(19) i=O

Equating terms with the same power of t in these equations, we arrive at the following recurrence formulae for ui + 2, wi + 2, and fi + 2 :

where

PO)

(21)

(22)

(23)

and we have all the Ui, Wi, and fi coefficients except for the first two, which should be found using the boundary conditions. The terms for Ui+2, wi+2, and fi + 2 converge to 0 as i+ cc. For this case we choose as degrees of freedom in the formulation the axial displacement, the lat- eral deflection, and the flexural rotation at the two ends of the beam element. At <=O we have

uo=u (0) (24)

wo=w (0) (25)

fo=f (0) (26) so the first three terms are readily known from the boundary conditions. The terms ul, wl, and fi are found as follows: all the ui’s, WI’s and fi’s are linearly dependent on the first two in each series, and we can write

u(1)=c$Lo+c2U* +&wg+C4w1

+Gfo+Gf1 (27)

w(1)=c~uo+c*u~ +cgwo+c~ow~

+Gfo+G2f1 (28)

f(l>=G,u0+C,4u, +clswo+c16w1

+G7fo+G8fi (29)

The eighteen C coefficients are functions of the axial, shear, and flexural stiffness of the element. C1 for example, is the value of u (1) calculated from eqns (14-16) using the recur-

268 M. Eisenberger; H. Abramovich, 0. Shulepov

rence formulae in eqns (20-22) for u. = 1 and u,=wO=w, =fo=fi =O. For the derivation of the stiffness matrix we have to apply unit displacements or rotation at each of the six degrees of freedom of the element, one at a time and cal- culate all the terms in the series for U, w and 4 using the recurrence formulas. Then the axial force, shear force, and the bending moment at the two ends of the element (t=O and t= 1) will be the stiffnesses for the member.

Thus, there are six sets of geometrical boundary conditions as follows:

(1)

(4

(3)

(4)

(5)

(6)

u(O)=l;

W(o)=f(o)=U(l)=W(l)=f(l)=o;

w(O)=l;

u(o)=f(o)=u(l)=W(l)=f(1)=o;

f (O)=l;

u(o)=W(o)=U(l)=W(l)=f(1)=o;

u(l)=l;

u(o)=W(o)=f(o)=W(l)=f(l)=o;

w(l)=l;

u(o)=W(o)=“f(o)=u(l)=f(l)=o;

f (l)=l;

u(o)=W(o)=f(o)=u(l)=W (l)=O;

Corresponding to these six sets there are six solutions ‘I)/~; i = 1, 6 for u(t), %‘i; i=l, 6 for w(Sy), and 3;; i=l, 6 for f (0 which are found using eqns (24-26) and (27-29). These are the dynamic shape functions for the laminated beam model as these are frequency dependent. Then, the holding actions, i.e. stiffnesses are:

A,, BII S(l,i)=-L~~i,,-L.Bi,,

S(2, i)= - 9 [%&, 1 -9-;,o]

(30)

(31)

(32)

k =0 J

S(6, i)= + i k.P,,k+% ,; kj)/i.k k-l L I

(35)

The natural frequencies of vibration for the member are the values of w that cause the

2 4 6 8 10 12

f [KHz1

Fig. 1. Determinant of the dynamic stiffness matrix vs. natural frequency for Example 1.

Table 1. Natural frequencies (in kHz) of a simply supported orthotropic (0”) graphite-epoxy beam (L=O-381

m, h=c=0*254)

Mode Ref. 4 Ref. 7 Present

1 0.755 0.755 0.755 106 2 2.543 2.548 2.547846 3 4.697 4.716 4.7 15963 4 - - 6.718828 5 6.919 6.960 6.959911 6 9.127 9.194 9.193958

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..L...... ..:.........; -5- : i ; ; .; : . . . . .._

-1l-l I I I I I I ._ 5 15 25 35

w



dynamic stiffness matrix for the element to become singular. A simple research routine is applied to find these values up to the desired accuracy.

EXAMPLES

Several examples will be given for verification of the results compared to the known values.7T4

.

50 100 150 200 250 300 Frequent y


The following AS/3501-6 graphite-epoxy material properties are used in all the examples: E, =14*5 x 10” N/m’, &=0*96 x 10” N/m*, G23=O*34 x 10” N/m”, G12=G13=0.41 x 10” N/ m*, v12=0.3, and p=157 x lo3 kg s* m-4. The shear correction factor k is taken as 5/6.

The first example is for an orthotropic (0’) beam.7,4 This problem was solved using the

-0.001 88.8 88.9 89

Frequency 89.1

Fig. 4. Determinant of the dynamic stiffness matrix vs. natural frequency for Example 3 - 5th and 6th frequen-

cies.

Table 2. Non dimensional frequencies (;=wLzL:p/EIh*) of (O/90/90/0) cross-ply beams (L/h=154

Beam

type 1 2

Mode

3 4 5 6

SS [Ref. 71 25023 8.4812 15.7558 23.3089 SS [present] 25023504 8.4812945 15.755931 17.259067 23.309265 CC [Ref. 71 45940 10.2906 16.9659 24.0410 31.2874 CC [present] 4594069 10.290759 16.966160 24.041380 31.287901 CF [Ref. 71 0.9241 4.8925 11.4400 18.6972 CF [present] 0.9241169 4.8925270 11.440113 17.259067 18.697446 CS [Ref. 71 3.5254 9.4423 16.3839 - 23.6850 CS [present] 3.5253917 9.4423758 16.384064 17.259067 23.685408

30.8386 30.839122

34.518135 26.2118 26.212145 31.0659 31.066347

Table 3. Non dimensional frequencies (~=oLz,~Z~/DII) of (O/90) beams with different boundary conditions (L/h=lO)

Mode Number

1 2 3 4 5 6 7 8 9

10

Fixed-Fixed Hinged-Hinged Fixed-Free

Ref. 9 Present Ref. 9 Present Ref. 9 Present

12.141 12.10808 8.1439 8.13392 2.2427 2.23948 28.473 28.37158 21.661 21.60865 12.494 12.46528 48.141 47.94420 43.788 43.64532 30.458 30.36466 69.449 69.14120 63.787 63.56258 50.765 50.61310 91.743 91.31179 89.150 88.92666 54.707 54.65661

102.66 102.56854 89.313 88.96338 74.216 73.91206 114.62 114.06012 114.30 113.76717 97.449 97.01444 137.38 136.67633 135.88 135.21732 120.26 119.70018 160.43 159.58815 159.97 159.14159 141.87 141.33001 175.55 175.05971 168.42 168.00770 146.95 146.67388

270 M. Eisenbergel; H. Abramovich, 0. Shulepov

present method and the results are compared with those from the literature.437 Figure 1 shows the plot of the determinant of the dynamic stiffness matrix for the beam vs. the natural

lr

I t

Modr 1 __.. _... . . . .

-u I I

0 : . 6 6 10

4 I

3- _I - ., Mode 3 :a . ..’

A- I 0 2 4 6 a IO

-2 : : - ._

.., i ._ .’

-3 I I I

0 I 4 6 6 10

-0 2 4 6 : 10

Fig. 5. Mode shapes for hinged-hinged beam of Example

frequency f in KHz as in the references.7Y4 This plot is a very efficient tool that enables us to identify the various kinds of modes that exist in the behavior of generally laminated beams:

._i Mode 6 L 1

0 2 4 6 1 10

‘., ,;

-1) I I 0 2 4 6 I 10

u displacement, ---- - w displacement, ----- - @ rotation, and ..... - 7 shear angle.


axially dominant mode, flexural dominant mode, and shear dominant mode. The first 6 natural frequencies for a short-thick (L/h= 1.5, h=l) simply supported beam are given in Table 1. As can be seen, in both7*4 the fourth frequency was skipped. This is the axial deformation mode.

The second example is for a symmetrically laminated beam as in Chandrashekhara et d7

In Figure 2 the determinant of the dynamic stiffness matrix is plotted vs. the non-dimensional frequency 0 for the clamped-simply supported boundary conditions. In Table 2, the first six non-dimensional frequencies of four layer symmetric cross-ply beams with different boundary conditions are presented. Here again, several of the frequencies are added to those given in Chandrashekhara et a1.7 We can see that these are not the high frequencies, but rather among the lower modes. These are the un-coupled axial vibration modes.

The third example is for a non-symmetric beam with two layers (O”/90”).9 The results for the first 10 natural frequencies of the beam, with 3 combinations of end conditions are compared with the results from Abramovich & Livshits’ in Table 3. For all these cases the axial displacements were restrained, so that the effect of the coupling is stronger. It can be seen that the results are in very good agreement for all frequencies and boundary conditions (with rela- tive differences of less than 0.5%). In Fig. 3 the determinant of the stiffness matrix for the hinged-hinged beam is plotted. Here we can see the first 16 natural frequencies. The 5th and 6th frequencies are very close, and in Fig. 4 one can see them in more detail. The first 10 mode shapes are given in Fig. 5. For each mode the axial displacement U, the transverse displacement w, the bending slope 4, and the shear angle y are plotted. All the mode shapes are normalized in such a way that the bending slope at the right end is unit. One can see that the 6th and 10th are the modes that are dominated by the axial deformations, and in all the modes and effect of coupling is very clear.

SUMMARY

In this paper the exact shape functions for the deflection and bending slope of composite laminated beam elements were used to derive the exact dynamic stiffness matrix for the beam. The element has only 6 degrees of freedom, as the classical frame element. It was shown that this method yields the exact results, and enables us to get all the natural frequencies. The advantages of the present method are in the ability to deal with general layouts and geometries of the structure and its boundary conditions at the ease of the general finite element method, using a minimal number of elements, but with exact results.

REFERENCES

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2.

3.

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Leissa, A. W., Recent studies in plate vibrations: 1981-1985. Part i: Classical theory. Shock Vib. Dig., 19 (1987) 11-18. Leissa, A. W., Recent studies in plate vibrations: 1981-1985. Part ii: Complicating effects. Shock I/ib. Dig., 19 (1987) 10-24. Vinson, J. R. & Sierakowski, R. L., The Behaviour of Structures Composed of Composite Materials. Martinus Nijhoff, Dordrecht, The Netherlands, 1986. Abramovich, H., Shear deformation and rotary inertia effects of vibrating composite beams. Comp. Struct., 20 (1992) 165-173. Teoh, L. S. & Huang, C. C., The vibrations of beams of fiber reinforced materials. J. Sound & Vibration, 51 (1977) 467-473. Chen, A. T. & Yang, T. Y., Static and dynamic formulation of symmetrically laminated beam finite element for a microcomputer. J. Comp. Mat., 19 (1985) 459-475. Chandrashekhara, K., Krishnamurthy, K. & Roy, S., Free vibrations of composites beams including rotary inertia and shear deformation. Comp. Struct., 14 (1990) 269-279. Singh, M. P. & Abdelnaser, A. S., Random response of symmetric cross-ply composite beams with arbitrary boundary conditions. AIAA J., 30 (1992) 1081-1088. Abramovich, H. & Livshits, A., Free vibrations of non-symmetric cross ply laminated composite beams. J. Sound & Vibration, 176 (1994) 597-612. Eisenberger, M., Exact static and dynamic stiffness matrices for variable cross section members. AiX4, 28 (1990) 1105-1109. Abramovich, H., Eisenberger, M. & Shulepov, O., Dynamic stiffness matrix for symmetrically laminated beams using a first order shear deformation theory. In ICCS 93, Madrid, Spain, July, 1993.

dynamic stiffness analysis of laminated beams using a first order shear deformation theory

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