Free Vibration of Composite Box Beams by Ansys

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12 INTERNATIONAL SCIENTIFIC CONFERENCE

1617 November 2012, GABROVO

FREE VIBRATION OF COMPOSITE BOX-BEAMS BY ANSYS.

M. Gkhan GNAY andTaner TIMARCI

Trakya University, Mechanical Engineering Department, Edirne, Turkey.

Abstract

Natural frequencies and mode shapes of CUS (circumferentially uniform stifnesss) and CAS (circumferentially

asimetric stifnesss) laminated cantilever box-beams are found by use of a finite element analysis package, ANSYS.

The applicability of ANSYS for analyzing the vibration of composite box-beams are validated by comparing thenumerical results with the ones obtained on the basis of the analytical and other finite elements techniques available in

the literature. The effects of the symmetries and the fibre orientation angle changes in the lamination on the vibration

characteristics of the beams are investigated.

Keywords: composite box-beams, vibration, ANSYS.

INTRODUCTION

The thinwalled box-beams made of the fiber-

reinforced composite materials are used

extensively in many engineering applications

because of their good mechanical properties,

such as high strength and stiffness to weight

ratios. Composite materials can be tailored to

provide the certain design requirements. There

are many articles published in order to

understand the dynamical behaviour of thecomposite box-beams. Chandra and Chopra

[1] investigated the vibration characteristics of

rotating composite box-beams. They solved

the governing equations by Galerkin Method

and compared the results with the ones of a

test facility for symmetric and antisymmetric

beams. Dancila and Armanios [2] presented

a solution procedure for the vibration problem

of the slender composite box-beams having

two types of configurations resulting (CUS

and CAS cases) in extension-twist andbending-twist couplings. The non-classical

effects are incorporated in the thin-walled box-

beams models by the approach of Qin and

Librescu [3]. They used Extended Galerkins

Method for the solution of the static and

vibration problem of the beams. Shamedri et

al. [4] also included the nonclasical effects in

their work on the investigation of static and

dynamic behaviour of composite box-beams.

Vo and Lee [5], based on the the classical

lamination theory, derived the equations of themotion of composite box beam for arbitrary

laminate confguration by Hamiltons principle

and they solved these equations by applying

displacement-based one dimensional finite

element model. As can be seen from the

previous works, the analitical approaches needto realize very cumbersome calculations

especially when the non classical effects are

considered. In this study, in order to

understand the capability of a finite element

package, ANSYS, it has been attempted to

obtain the natural frequencies and mode

shapes by use of this package. The numerical

results were compared with the ones available

in the literature.

THE FORMULATION

The geometry of the box-beam and the

coordinate system is shown in the Figure 1

while Figure 2 shows CUS (circumferentially

uniform stifnesss) and CAS (circumferentially

asimetric stifnesss) configurations. In general,

two coordinate systems which are related toeach other, are required in order to describe

the behaviour of the beam (Fig.1). For

comparison reasons, the model presented in

the work of Qin and Librescu [3] is presented

briefly in this section. In analysis, it is

assummed that the cross-section of the beam

does not deform in its own plane and the

deformations are small. It may be referred for

the other assumptions to the previous articles

[3,5] depending on the inclusions of the non-

classical effects, such as transverse shear.


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Fig. 1. The geometry of the beam and coordinatesystems

Lay-Up Flanges Webs

Top Bottom Left Right

CAS *+6 [-+6 */-+3 */-+3

CUS *+6 *+6 **+6 *+6

Fig. 2. Symmetric and antisymmetricconfigurations

The displacement field as given in [3] is:

(1)

Warping function Fw is defined again in Ref

[3-4]. (.) and (.),s denote the derivatives with

respect to z and s respectively. After having

applied Hamiltons Principle, the governing

equations for CAS case where they involve

three unknowns, v0, , x, corresponding

to the vertical bending, twist andtransverse shear, from Ref. [4] are as follows:

Here, global stifness coefficients aij and

inertia coefficients bi are given in Ref. [4]. Inthe study mentioned, although state ofstress can be split into vertical bending,twist, transverse shear, there are couplings ofextension-twist and bending-shear in CUS

case while in CAS case extension-transverse

shear and bending-twist couplings, in general.

SOLUTION BY ANSYS

ANSYS is a finite element modeling package

for numerically solving various structural

problems. The more details and tutorials can

be found in [6,7] In the study presented, thenatural frequencies and mode shapes are

obtained for cantilever beams for antisymetric

and symmetric configurations by use of

ANSYS. In the analysis SHELL281 8-node

element was used. In this modeling, the first

order shear deformation theory is adopted for

layered composite structure applications. It

may be referred to Ref.[8] for different shear

deformable models. The numerical results arecompared with the ones presented in the works

of Chandra and Chopra [1] and Qin and


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Librescu [3] where the solutions are obtained

by experimental and numerical methods,

respectively. The same results are also

available in [5]. In Table 1 the configurations

corresponding to the different lay-ups are

presented. The values of the natural

frequencies obtained for these certainconfigurations and the relative errors with

respect to the ones of the previous works are

given in Table 2. As seen from this table the

results give very reasonable values especially

for the compared numerical values obtained

numerically in Ref [4], by Extended

Galerkins method except for the case CAS2

for which there is a difference about 9.5 %.

Table 1. Material properties for CAS and CUS

cases.

Table 2. Geometrical properties of the Box-beams

Table 3. The lay-ups used for comparison withprevious works.

We can say that the ANSYS results are

generally lower than the experimental ones,

except for the CUS2-3beams. After having

validated the results obtained by ANSYS, in

Table 5, the variation of the first two natural

frequencies corresponding to the x and y

directions with respect to the fibre orientation

angle for CAS lay-up is presented.

Table 4. The comparison of the frequencies (Hz)for the certain lay-ups and modes with previousworks.

Mode Ref[3] Ref[4] ANSYS Er(%) Er(%)

Ref[3] Ref[4]

CAS2 y1 20.96 21.80 19.73 -5.87 -9.50

y2 128.36 123.28 123.32 -3.93 0.03x1 38.06 - 37.53 -1.39 -

CAS3 y1 16.67 15.04 14.58 -12.54 -3.06

y2 96.15 92.39 91.23 -5.12 -1.26

x1 29.48 - 25.01 -15.16 -

CUS1 y1 28.66 30.06 28.37 -1.01 -5.62

CUS2 Y1 30.66 34.58 34.29 11.84 -0.84

CUS3 y1 30.00 32.64 32.35 7.83 -0.89

R-Er %=(ANS-Ref)]X100/Ref. %

Table 5. The natural frequencies for CAS beam

with respect to the fiber angle change.

Fiber Angle () y1 (Hz) y2 (Hz) x1 (Hz) x2 (Hz)

0 43,246 256,430 68,075 402,570

10 35,393 216,600 61,538 372,250

20 26,001 161,860 49,969 307,890

30 19,731 123,320 37,530 233,150

40 15,834 99,050 28,133 175,320

50 13,628 85,256 22,722 141,730

60 12,440 77,801 20,000 124,750

70 11,827 73,931 18,729 116,780

80 11,542 72,078 18,192 113,390

90 11,459 71,484 18,046 112,450

As seen from this Table, the fundamental

frequency belongs to the bending mode in

vertical direction, y. Table 5 and 6 show the

effect of the increase of number of the layers

by keeping the layer thickness constant on the

frequencies for CAS2 and CUS beams

respectively.

Table 6. The natural frequencies for CAS (=300)

beam with respect to the number of layers.

Layer

Count

(2n)

y1 (Hz) y2 (Hz) x1 (Hz) x2 (Hz) 0 (Hz)

4 19,692 123,040 37,513 233,020 680,100

6 19,731 123,320 37,530 233,150 693,326

8 19,759 123,510 37,541 233,230 698,429

10 19,762 123,540 37,376 232,130 687,940

12 19,724 123,300 37,039 230,000 678,393

14 19,681 123,030 36,668 227,690 670,297

16 19,645 122,800 36,307 225,460 633,610

18 19,617 122,630 35,972 223,380 658,170

20 19,600 122,520 35,666 221,490 653,799

As can be observed from these tables, as thenumber of layers are increased the values of

E11=141.96 GPa E22= E33=9.79 GPa

G23= 4.83 GPa G12= G13= 6 GPa

12= 13= 0.42 , 23=0.5, =1445 kg/m3

L (length) 762 mm

2b (Outer width) 24.2 mm

2a ( Outer depth) 13.6 mm

h (thickness) 0.762 mm

2n ( number of layers) 6

( Layer thick.) 0.127 mm

Lay-Up Flanges Webs

Top Bottom Left Right

CAS2 [30]6 [-30]6 [30/-30]3 [30/-30]3

CAS3 [45]6 [-45]6 [45/-45]3 [45/-45]3CUS1 [15]6 [15]6 [15]6 [15]6

CUS2 [0/30]3 [0/30]3 [0/30]3 [0/30]3

CUS3 [0/45]3 [0/45]3 [0/45]3 [0/45]3


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1

MN

MX

X

Y

Z

.420E-07

.8555531.711

2.5673.422

4.2785.133

5.9896.844

7.7

SEP 18 2012

11:16:26

NODAL SOLUTION

STEP=1

SUB =1

FREQ=19.731

USUM (AVG)

RSYS=0

DMX =7.7

SMN =.420E-07

SMX =7.7

1

MN

MX

X

Y

Z

.233E-06

.8555611.711

2.5673.422

4.2785.133

5.9896.844

7.7

SEP 18 2012

11:16:57

NODAL SOLUTION

STEP=1

SUB =3

FREQ=123.322

USUM (AVG)

RSYS=0

DMX =7.7

SMN =.233E-06

SMX =7.7

0 10 20 30 40 50 60 70 80 9010

20

30

40

50

60

70

Fiber Angle ()

x

1

(Hz)

CAS

CUSfrequencies decrease slightly for CAS case whilethey increase for CUS case.

Table 7. The natural frequencies for [30]2n (CUS)beam with respect to the number of layers.

Layer

Count(2n)

y1 (Hz) y2 (Hz) x1 (Hz) x2 (Hz) 0 (Hz)

4 18,225 113,780 28,786 178,830 510,925

6 18,240 113,900 28,795 178,900 516,271

8 18,260 114,040 28,806 178,980 519,334

10 18,285 114,200 28,819 179,070 522,070

12 18,314 114,390 28,834 179,180 524,850

14 18,349 114,610 28,850 179,300 527,792

16 18,387 114,850 28,869 179,430 530,919

18 18,430 115,130 28,889 179,570 534,222

20 18,477 115,420 28,910 179,720 537,680

In Fig.3 the output for the first eightfrequencies of CAS2 lay-up is presented.

Fig. 3. The ANSYS output of the naturalfrequencies for CAS2 beam.

In Fig.4. the variations of the natural

frequencies of CAS and CUS cases

corresponding to the x-direction, with respect

to the fiber orientation angle are shown. As

shown from the figure, the frequencies

decrease sharply from 00 to 600 while the

values of CUS case are higher than the ones of

CAS case.

Fig. 4. The change of the frequencies with respectto fiber angles for CAS and CUS cases

Figures 5 and 6 are the mode shapes of the

box-beam corresponding to the first and

second modes in y-direction respectively. As

shown from the figures, there is also

extensional displacement in z-direction

coupled to the displacement in y-direction.

Fig. 5. The mode shape fory1= 19.73 Hz of

CAS2 beam.

Fig.6. The mode shape fory2= 123.32 Hz ofCAS2 beam.


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CONCLUSION

This is a preliminary study to understand the

dynamic behavior of composite box-beam

realized by use of ANSYS finite element

package. It can be observed from the results,

ANSYS results are in quite reasonable limitscompared with ones of previous works. The

study based on the ANSYS can be extended to

the static analysis of composite box-beams with

different boundary and loading conditions.

REFERENCES

1-Chadra R., Chopra I. Experimental-

Theoretical Investigation of the vibration

characteristics of rotating box-beams. J. of

Aircraft, 29:4, 657-664, 1992.

2- Dancilia DS, Armanios EA. The influence

of coupling on the free vibration of anisotropic

thin-walled closed section beams. Int J Solid

structures 35:( 23) 3105-3119, 1998.

3-Qin Z, Librescu L. On a shear- deformable

theory of anisotropic thin walled beams:

further contribution and contribution.

Composite Structures 56:345-358, 2002.

4-Shadmehri F, Haddadpour H,

Kouchakzadeh MA Flexural-torsional behaviorthin walled composite beams with closed

cross-section, Thin-Walled Structures 45: 699-

705, 2007.

5- Vo TP, Lee J. Free vibration of thin-walled

composite box beams. Composite Structures

84:11-20, 2008.

6-http://www.ansys.com/

7- www.mece.ualberta.ca/tutorials/ansys

8- Karacam F, Timarc T. Bending of cross-

ply beams with different boundary conditions.

UNITECH-05, Gabrovo, Bulgaria. Proceedings

of Int. Scient. Conference II:137-142, 2005.

.
http://www.ansys.com/http://www.ansys.com/http://www.ansys.com/

Free Vibration of Composite Box Beams by Ansys

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