ELSEVIER 0263-8223(95)00023-2

Composite Stncctures 32 (1995) 3-11 0 1995 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0263-8223/95/$9.50

Dynamic analysis of open cracked laminated composite beams

S. M. Ghoneam Department of Production Engineeting & Mechanical Design, Faculty of Engineeting, Menoujia University, Shebin-El-Kom,

Egypt

Composite beams of various orientations with fiber reinforced plastics find increasing application in structures nowadays, for their light weight, high strength and stiffness, and high damping characteristics. Determination of the dynamic characteristics of cracked laminated composite beams (CLCB) is essential not only in the design stage but also in the performance of the structure system, since the presence of cracks, with various types of fiber orientation, and boundary fixations of the beams has a dominant effect on the dynamic nature of the system.

In the present paper, the dynamic characteristics laminated composite beams (LCB) with various fiber orientations and different boundary fixations have been presented and discussed in the absence and presence of cracks. A mathematical model is developed, and experimental analysis is utilized to study the effects of different crack depths and locations, boundary conditions, and various code numbers of laminates on the dynamic characteristics of CLCB. The analysis shows good agreement between experimental and theoretical results.

INTRODUCTION

Dynamic analysis of the laminated composite beam is one of the most serious problems in machine element structures. The analysis of these elements and their dynamic characteristics is of great interest due to its practical impor- tance. Cracks which appear in composite structures, boundary fixations, and code num- bers change their dynamical characteristics due to a change in its flexibility. The effects of cracking on the behaviour of a structural beam have been the subject of several investigations. The cracks were modeled by a flexible element whose stiffness depended on a simple stress- strain problem. This model is sufficient for simple stress-strain problems. The effect of cracks on the dynamic behavior of a cracked beam was investigated for an isotropic mate- ria1.2-7 Lim & Tay introduced a constitutive model of the damage state for composite lam- inates, based on strain energy, to predict the stiffness loss due to matrix cracking in cross-ply laminated composite plate. The finite element method utilizing conventional plane stress finite

3

elements is also applied to laminated beams9-14 with separate elements for each lamina. In Ref. 15, an example of the dynamic analysis of a laminated composite cantilever beam of three layers was discussed and investigated.

The present paper presents a numerical and experimental analysis of eigen parameters on a laminated composite beam with various orienta- tions, carried out for different boundary fixations, and in the absence and presence of cracks. A mathematical model is developed which represents the CLCB. The model takes into consideration the effect of crack location and size, fiber orientation, and boundary fixa- tion. The experimental work is carried out on five specimens with five layers of composite laminated beam with different fiber orientation and various boundary fixations. These speci- mens have been manufactured using hand layout techniques. The experimental tests are carried out by using a hammer test and fre- quency response function is displayed on an FFT analyzer. The comparison between the experimental and numerical results are investi- gated, and the tight connection between them is shown.

4 S. M. Ghoneam

NUMERICAL PROCEDURE

Because of discontinuity of deformation in the crack element, a suitable shape function to express the elastic potential energy and kinetic energy is difficult. Calculation of the additional stress energy of a crack, however, has been studied in fracture mechanics and the flexibility coefficient expressed by the stress intensity factor can be derived by mean of Castiglianos theorem in the linear elastic range.16

In Fig. 1 (a) the CLCB is divided into elements. The behaviour of the element right of the cracked element may be regarded as external forces, while the elements situated in its left as constraints. From the equilibrium condition, the stiffness matrix of the cracked element with specified constraints may be calculated as follows.

The strain energy of an element without a crack, ignoring shearing action, is:

V0=(M2L +A4PL2 +P2L3/3)/2(E *I).

The contribution of strain energy due to the crack in the absence of axial force is:

V,=b a s

{[(K,,+K,,)2+K~p]IE*} da 0

where

K,,=(6Mlbh2) fir;,(s),

K~p=(3PWz2) JG F,(s),

K2p=(plbh) dz 25(s)

(1)

(2)

F,(s)=4%)$(742) 0.923 + 0.199 [ 1 - sin ( 7r.r/2)14

cos(ns/2) 9

and

&(S)=(3S-2S2) 1.122 -0.561s + 0.085~~ + 0.18~~

JT-S

In the absence of a crack, the flexibility coefficient for an element is

clzj= t12V,

8PiaPj , P1=P, P,=M, i,j=1,2 (3)

Scc.A-A

0

Fig. 1. Schematic diagram of CLCB.

Cracked laminated composite beams

and the additional flexibility coefficient due to a crack is:

a*v, a?i= aPiaPl 7 P1=P, P2=A4, i,j=1,2 (4)

The total flexibility coefficient is:

Cli,j=ClEj + Cfl,j i,j=l, 2

From the equilibrium condition (Fig. 1 (b))

[Pi,MiPi+lMi+I]T=TIPi+lMi+l]T

where

(5)

(6)

L

-1 -L 1 OT T=

0 -1 0 1 1 By the principle of virtual work, the stiffness matrix of the cracked element may be expressed as:

K,=TTa-T. (7)

Suppose the crack affects the stiffness only. The difference between K, the stiffness matrix of the CLUB, and K, the stiffness matrix of the LCB, is matrix K,.

The stiffness matrix of the LCB is given by

4446 *A 83G *A -39G *A -7G*A

12OL 120 12OL

7E I 16LG *A 7G*A -+

3L 120 120

4446 *A

120

E*A 4LG*A

3L- 120

-83G *A

12OL

symmetric

where (Fig. 1 (c))

120

7E *I 16LG *A -+

3L 120

(E*I)= f: 2E*b dk+lyzdy+_ s

i (E*)k (d:+l-d:). k=O 4 k=O

Referring to Ref. 17, the E * and G * are given as:

l=c+(& _ 2) CSS*+$; and E El1

1 -=4

1+2v1* 1 +

G* El, %-

where C=cos 0, and SS=sin 0.

(8)

(9)

(10)

6 S. M. Ghoneam

By applying the mixture rule the elastic moduli of E-glass/polyester lamina are computed. The volume fraction of fibers (I$) was obtained by a firing process and was found to be O-50. These properties are listed in Table 1.

The inertia matrix, m, of the CLCB in the same form as the inertia matrix of the LCB, is given as:

r 128A 19A PL 2441 0 -561

m 4x4- -- 1680 128A 0

1 symmetric 2241 1 where p is the density; its value is given in Table 1.

The eigenfrequency can be evaluated from the solution of the characteristic equation directly. The characteristic equation for an undamped LCB may be evaluated as:

The characteristic equation for the CLCB is:

([E] -iim) [z&l = TO]. (12) From eqns (11) and (12), the eigen frequencies have been determined for different boundary fixations, and various crack depths and locations.

Table 1. Elastic moduli of E-glass polyester

Elastic modulus El1 @Pa) EZZ @Pa) GIz Pa) v12 P (Kg/m3)

Results using mixture rule 36.75 6-67 34 0.26 1750

EXPERIMENTAL PROCEDURE

The test specimen is a beam 300 x 25 x 3 mm. Five specimens with five layers were constructed and manufactured using a hand layout tech- nique, with fiber orientation sequences [0]5, [30]5, [45]5, [60]5, and [90]5. Four boundary fixations for the LCB were taken into account as clamped-clamped, clamped-simple support, simply supported, and clamped-free. The first four frequencies are measured for different cases of boundary fixations.

In the presence of a crack, the crack was initiated for each CLB with a coping saw cut. It was propagated successively with 2-5 mm steps, and the values of a/h equal to O-1, O-2, O-3, 0.4, 0.5, and 0.6 were considered, where a is the crack depth. The crack locations are chosen at Z/L equal to 0.0, 0.25 and O-5 of the CLCB from the left end of each specimen, where 2 is the length of the beam segment to the left of the crack site. Due to the difficulty of specimen

production, the measurements were carried out for specimens [0]5, [30]5, and [60]5 with clamped-clamped fixation, and specimen [0] 5 for clamped-free fixation.

The experimental set up is shown in Fig. 2. The specimen is located in a test rig and excited by impact hammer (type 8202). This resembles an ordinary hammer but has a force transducer (type 8200) built into the tip to register the force input used to excite the sample at mid- point position. The charge amplifier (type 2635) is used to generate the signal from the hammer to the dual channel analyzer (type 2034) at A. The vibration response is registered by a suitable piezoelectric accelerometer (type 4374, weight 2.4 g). The vibration meter (type 2511) is utilized in connection with the accelerometer to generate the signal to the dual channel analyzer (type 2034) at B. The frequency response spec- trum can be obtained from the printer which is supported by the desk top computer and the dual channel analyzer.

Cracked laminated composite beams

Printer

Cl&B Ii

Ged purpose vibration meter

2511 lmpactbammer / 8202

/. I_ ,$

/ ,I \ 5 Accelerometer

43 14

Q \

!I ? 1

Fig. 2. The experimental setup.

RESULTS AND DISCUSSION

In the absence of cracks in the CLB, the reso- nance frequencies of the CLB of size 300 x 25 x 3 mm have been recorded and ana- lysed for different fiber orientations and boundary fixations at the same five layers. As a sample of the experimental results, the fre- quency response spectrums for the CLB with orientation [45]5 and various boundary fixations are shown in Fig. 3. The measured and com- puted first four frequencies are given in Table 2. The experimental results verified the theoret- ically obtained values from eqn 11. The comparison between these results indicates a good agremeent between them. It can be seen that the frequencies of specimen [90]5 are lower than those of the other specimen, and specimen [0]5 has higher ones. The natural fre- quencies in the specimen [0]5 are 2.3 times greater than those in specimen [90]5. Changing the fiber orientation of the CLB from [0]5 to [30]5 decreases natural frequencies by 40% approximately. The changing of the fiber ori- entation angle in specimen [30]5 and specimen [45]5 has an effect on the natural frequencies of about 18%, for specimen [45]5 and specimen [60]5 the effect is about 9%, and for specimen [60]5 and specimen [90]5 the effect is about 3%. From the Table it can be noted that increasing the stiffness of the CLB due to the end fixation and fiber orientation has a promi- nent influence in increasing the frequencies. For

instance, the specimen [0]5 with clamped- clamped ends fixation has higher natural frequencies than the specimen [90]5 with clamped-free ends fixation. Thus the frequency level may be controlled by changing the fiber orientation and boundary fixation, and conse- quently the results obtained are useful for the designer in order to select the proper fiber ori- entation and boundary fixation.

In the presence of cracks (CLCB), experi- mental measurement has been carried out to verify the developed finite element model of CLCB. Comparisons between the experimental and the finite element model results of the specimens [0]5, [30]5, and [60]5 for clamped- clamped fixation are presented. Moreover, the experimental and theoretical analysis of speci- men [0]5 for clamped-free is investigated. From eqn 12, the eigen frequencies have been deter- mined for different crack depths and locations for the above mentioned fiber orientation angle and boundary fixation. The experimental and computed results are shown in Figs 4-6. The computed values are represented by the continuous lines, and the experimental meas- urements are represented by the symbols. The variation in the natural frequency is defined as the differences between the natural frequency of the CLB, a, and the natural frequency of the CLCB, o, i.e. Aw=o--0,. The normalized eigen frequency due to the crack is do/o. Gen- erally, it is evident that the calculated values agree with the experimental results. Figure 4

8 S. M. Ghoneam

12 Freq resp HI MAG Input Main Y: 5.6dB

Y: 20.0 dB 80 dB x: 76 Hz x: 0 Hz + 3.21 Hz LIN

# A: 20

(a) Clamped-Clamped

W12 Freqnsp y: 9.4 dB

Hl 8odB

MAG Input Main Y: -13.8dB

x: 56Hz x: OHz+ 1.61 Hz LJN

#A:20

(b) Clamped-simple supported

W12 Freqresp Hl MAG Input Main Y: -0.1 dB

Y: 9.4 dB 80 dB x: 36 Hz x: OHz+ 1.6kHz LIN

#A:20

(c) Simply supported

12 Y:

Fres =P 20.0 dB

Hl 8OdB

MAG Input Main Y: 9.9dB

x: 12Hz x: 0 Hz + 3.2 kHz LlN

#A:20

(d) Clamped-free

Fig. 3. Frequency response spectrums for CLB with orientation [45]5 and different boundary fixations (fundamental frequency).

Cracked laminated composite beams 9

Table 2. Numerical and experimental values of the first four frequencies in Hz for various laminated codes and different boundary fixation of CLB

Boundary fixation* c-c C-S s-s C-F

Lamin. codes Freq. no. Th. Exp. Th. Exp* Th. Exp. Th. Exp.

PI5 : 3 4

[3015 1

32

[4515

WI5

[9015

4

1 2 3 4

1

; 4

: 3 4

157.0 162 108.2 110 432.8 433 350.7 355 847.6 862 731.6 736

1402.7 1412 1249.0 1255

93.5 92 64.4 257.6 262 208.7 504.7 510 435.5 834.9 838 743.4

2:; 440 747

76.6 76 52.8 56 211.1 212 171.0 174 413.4 416 356.9 360 684.2 675 608.8 612

69.8 192.4 376.9 623.5

72 194 385 630

48.1 51 155.8 158 325.2 330 555.1 564

67.1 68 46.3 47 184.9 185 149.9 155 362.3 368 312.7 318 599.2 608 533.7 540

69.3 70 24.7 276.8 278 154.7 622.7 628 433.0

1106.9 1115 848.4

41.2 44 14.7 164.8 166 92.2 370.7 375 257.8 658.9 660 505.2

33.8 36 12-o 134.9 136 75.5 303.9 306 211.3 540.0 545 414.2

30.8 32 11.0 123.0 125 68.8 276.8 278 192.5 492.0 500 377.4

29.6 32 10.5 118.3 120 66.1 266.1 272 185.0 473.0 482 362.5

26 160 441 858

14 94

265 508

:: 220 418

10 70

190 385

10 67

192 365

*C=Clamped, S=Simple-supported, F=Free.

shows the variation of the first three eigen fre- quencies as a function of crack depth for the three values of crack locations for specimen [0]5 with clamped-clamped end fixation. It is clear that the fundamental frequency, in the presence of a crack, decreases with an increase in crack depth. For a given crack depth, the fundamental frequency decreases as the crack location nears the middle of the beam. For the second and third natural frequencies, the gen- eral trend of a decrease in the natural frequency with an increase in crack depth is observed. However, due to the nodes of the beam for the second and third mode shapes, the change in the natural frequency with the crack location is not as monotonic as in the first mode but depends on how close the crack is to the mode shape node. The first three eigen fre- quencies for the specimen PI5 with clamped-free fixation, and crack location at the middle of the beam, is shown in Fig. 5. It can be noted that the rate of change in the first fre- quency is largest, followed by the third frequency and second frequency respectively, this is due to the crack hearing the mode shape node. The relationship between the normalized frequency for the first frequency (Ao/al) and

(a/h) for various fiber orientations is shown in Fig. 6. It can be noted that increasing the fiber orientation angle increases the rate of variation frequency, as the fiber stiffness decreases with increasing fiber orientation angle.

CONCLUSIONS

The dynamic analysis of laminated composite beams with various fiber orientations and differ- ent boundary fixations in the absence and presence of cracks is investigated analytically and experimentally. The fiber orientation and boundary fixation of the CLB has a significant influence on the dynamic properties of the CLB, depending on the type of fiber and matrix material.

The element stiffness matrix of a CLCB has been derived from an integration of stress fac- tors. A finite element approach for dynamic analysis of a CLCB has been proposed. Simple and convenient calculation is its distinguishing feature. The specimens of orientation sequences [0]5 and [90]5 have a higher and lower fre- quencies respectively compared with the specimens of the other orientation sequence

10 S. M. Ghoneam

v.-

0 0.1 0.2 0.3 0.4 0.5 0.6

Oil 0.3

t zJL=o I

f 0.2

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

Fig. 4. Variation of the first three eigen frequencies with different crack depth of different CLCB geometril c ratio

(clamped-clamped, [0]5).

with the same boundary fixation, because its ori- entations sequences are expected to make the CLB more stiff and flexible respectively.

The increasing of the fiber orientation angle decreases the eigen frequencies of the CLCB, because the stiffness of the CLCB is inversely proportional to the increase of the fiber orienta- tion angle.

Pig. 5. Variation of the first three eigen frequencies with different crack depth at Z/L=05 for specimen 1015

- (clamped-free). -

16015

0.3 -

<

d

0 0.1 0.2 0.3 0.4 0.5 0.6

dh

Fig. 6. Variation of the first eigen frequency with differ- ent crack depth for different fiber orientations for

clamped-clamped at Z/L =O.

The presented procedure can be used to identify cracks by linking the variation in service of the composite structural beam natural fre- quencies to the structural changes due to the cracks.

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