Studying the Effect of Fiber Orientation and Skew Angle on

Journal of Mechanical Engineering Research and Developments

ISSN: 1024-1752

CODEN: JERDFO

Vol. 43, No. 7, pp. 199-212

Published Year 2020

199

Studying the Effect of Fiber Orientation and Skew Angle on The

Fundamental Frequencies of Simply Supported Composite Plates

Using Finite Element Method

Raghad Azeez Neamah*†, Zainab M. Shukur†,‡, Luay S. Al-Ansari†, K. D. Dearn‡

† Mechanical Engineering Department – Faculty of Engineering – University of Kufa, Iraq 1AL-Mason ‡ Institute of Tribology, Department of Mechanical Engineering, University of Birmingham, Edgbaston,

Birmingham, B15 2TT, UK.

*Corresponding author E-mail: [email protected]

ABSTRACT: Due to high strength to weight ratio and high specific modulus, laminated composite plates are

different applications, the skew laminated composite plate are widely used in these applications. Therefore, the

free vibration behavior of skew composite plate analysis is one of the most important behavior and it is the main

purpose of several studies. In this work, two finite element models were built to simulate the skew composite

plates using ANSYS APDL version (17.2). The first model was the 2-D models based on SHELL281 while the

second model was 3-D models based on SOLID186. The two models were used to study the effects of skew angle,

width-to-length ratio, fiber orientation, supporting type and layers layout on the fundamental frequency of

laminated skew plates. The skew angle was varied from (0o) to (75o) while the fiber orientation was varied from

(0o) to (90o) and the width-to-length ratio was (0.5,1,1.25, 1.5 and 2). (SSSS), (SFSF) and (FSFS) types of

supporting were selected and three types of layers layouts were used. The results showed that skew angle and

width-to-length ratio fundamental frequency are strongly and separately effect on the fundamental frequency.

While the fiber orientation, supporting type and layers layouts have a slight effect on the fundamental frequency.

Finally, the combination effects of the five parameters gave a different variation form in the fundamental

frequency of the laminated skew plates.

KEYWORDS: Free Vibration, Skew Plate, Frequency Parameter, Length-to-Width Ratio, ANSYS Software,

Composite Plate, Finite Element Method.

INTRODUCTION

Due to high strength to weight ratio and high specific modulus, laminated composite plates are various types of

application such as air and space craft, marine vehicles, fuselage panels, nuclear structures, turbine blades and

automobile body panels. In modern era of composites, skew laminated composite plates are also being

increasingly used, for example in the wings of aircraft. Generally, the analysis of skew composite plate for free

vibration behavior is difficult or not available for different reasons like: (a) the material properties of composite

material are not modeled accurately and when accurate model is used, the material constant will increased, (b)

In laminated composite plate, the number of layers is the main challenge of any mathematical analysis. Therefore,

the approximate or numerical methods are used for research the free vibration behavior of skew composite plates.

The free vibration of skew isotropic and laminated composite plates was extensively studied by many researchers

over the past few decades using different methods like Rayleigh-Ritz method and Finite Element method [1-28].

For isotropic plates, Lee and Han studied free vibration behavior based on the theory of first order shear

deformation and using a shell part degenerated by nine-nodes [29]. Their shell model gave significant

implementation advantages since it consistently uses the natural coordinate system. They carried out numerical

examples then compared their results with the existing exact solutions. Hossain et al. developed a new finite

element model based on the theory of first-order shear deformation, for modeling the anisotropic and laminated

doubly curved composite shell panel with moderately thick curvature [30]. Their results were compared with the

available numerical and analytical results and they found an excellent agreement in shallow and deep shells case.

They carried out numerical examples to explain the element performance. Garg et.al. investigated the free

mailto:[email protected]

Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using

Finite Element Method

200

vibration problem of four types of skew plates ((a) isotropic plate, (b)orthotropic plate, (c) layered anisotropic

composite plate, and (d) sandwich skew laminates composite plate) by utilizing isoparametric finite element

model [31]. They compared their results with available results in the literature to check the model accuracy. They

carried out the natural frequencies of cross-ply and angle-ply skew laminates plates. They introduced a new result

on skew sandwiches for different lamination parameters and skew angles. Liu et al. analyzed the static behavior

and the free vibration behavior of general shell structures by using the element-free Galerkin method [32].

Pradyumna and Bandyopadhyay used a higher-order finite element to analyze the free vibration problem of FG

curved panels [33]. Srinivasa et. al. has analyzed the flexural natural frequencies of isotropic and laminated

composite skew plates with clamped and clearly assisted limit conditions [34]. with the methodology of finite

elements. They used two types of elements (CQUAD4 element and CQUAD8 element) to study the effects of (1)

aspect ratio, (2) length-to-thickness- ratio , and (3) the skew angle plates on the natural frequency of isotropic

skew plate. For skew composite plate , they studied the effect of five parameters (1) aspect ratio, (2) length-to-

thickness- ratio, (3) the skew angle (4) fiber orientation angle and (5) numbers of layers on the free vibration

behavior. They found that frequencies increase when the skew angle increases and the variation of frequency with

the number of layers is not appreciable if they used large number of layers.

The free vibration problem of rectangular composite laminate plate for various boundary conditions was studied

by Deepesh Bhavsar using ANSYS APDL software. He studied the effects of (1) side-to-thickness ratio, (2) aspect

ratios, and (3) ratio of Modulus of Elasticity on the first five natural frequencies of plate for different boundary

conditions [35]. The free vibration problem of rectangular composite laminate plate for various boundary

conditions was studied by Deepesh Bhavsar using ANSYS APDL software. He studied the effects of (1) side-to-

thickness ratio, (2) aspect ratios, and (3) ratio of Modulus of Elasticity on the first five natural frequencies of plate

for different boundary conditions [36]. Khalafia and Fazilatib investigated the free vibration analysis of finite

square and skew laminated plates by developing a "NURBS-based isogeometric finite element formulation" [36].

They assumed variable stiffness plies to employ the curvilinear fiber reinforcements. The cubic NURBS basis

functions are employed to approximate the geometry of the plate while simultaneously serve as the shape functions

for solution field approximation in the analysis. They studied the accuracy and effectiveness of the suggestion

method by comparing their results with results in the available literature.

Reeta Bhardwaj and Naveen Mani studied the vibrational behavior of parallelogram skew plate with non-

homogeneous material [37]. They assumed 1D variation of thickness and 2D variation of temperature on clamped

edges. They used Rayleigh Ritz method to solve governing differential equation of motion for vibration behavior.

The governing differential equation of motion for vibration analysis is solved by using Rayleigh Ritz technique

and time period is calculated for the combination of different variation of plate parameters. The results show that

the suggested model offers a good appropriate data for time period of frequency which will be helpful for structural

design. The linear theory for moderately thick structures and strong formulation finite element method (SFEM)

were used by Nicholas Fantuzzi et. al. to calculate the first three natural of laminated composite plates frequencies

[38]. Isanaka et. al. used the finite element method to investigate the free vibration of simply supported stiffened

plates with three different shapes (a) circular, (b) square and (c) rectangular (with different aspect ratio) [39]. They

studied the effects of skew angle and the stiffener properties on the free vibration problem. Their results showed

that the stiffness increases with increase in skewness of the geometry in bare and stiffened plates bare and stiffened

plates. In rectangular plate, the maximum overall stiffness was found at aspect ratio 1.75, while in circular plate,

the maximum overall stiffness was found in the case of bare plates.

From previous literatures, the researchers studied the effects of (1) aspect ratio, (2) length-to-thickness- ratio, (3)

the skew angle and (4) numbers of layers on the free vibration behavior using different numerical methods

(specially finite element method). In this work, the behavior of free vibration skew composite plates was studied

by commercial finite element software (ANSYS APDL software version 17.2). Three types of laminated

composite plates wear considered depending on the layers arrangement (or layers layout) and these types were

(0/β/0), (β/0/ β) and (β/β/ β). Also, three types of supporting plate were used in this work to investigate the effect

of layers layout, fiber orientation, skew angle and the length-to-width ratio on the fundamental frequency.

PROBLEM DESCRIPTION AND STUDIED CASES



201

In this work, the three types of laminated composite plate were considered, and these types depends on the layers

arrangement (or layer layout) and these types are (0/β/0), (β/0/ β) and (β/β/ β). Three types of supporting (i.e.

boundary conditions) were simply support at all edge (SSSS) , simply support at left and right edges and free at

top and bottom edges (SFSF) and simply support at top and bottom edges and free at left and right edges (FSFS).

The geometry, dimensions and layers layout of skew composite plate is illustrated in Figure (1) and Table (1).

The skew angle (θ) is (0o, 15o, 30o, 45o, 60o and 75o). The physical and mechanical properties of the layer of

composite materials are listed in Table (2).

(0/β/0)

(β/0/β)

(β/β/β)

Geometry and Dimensions. Layout of Layers

Figure 1. Geometry, Dimensions and Layouts of Skew Plates Used in this Work.

Table 1. The Dimensions of the Plates Used in this Work.

No. Width (a) (m.). a/b Ratio. Length (b) (m.).

1 0.3 0.5 0.6

2 0.3 1 0.3

3 0.3 1.25 0.24

4 0.3 1.5 0.2

5 0.3 2 0.15

Table 2. The Physical and Mechanical Properties of the Composite Materials Used in this Work.

No. Property Symbol Value Unit

1 Longitudinal Modulus of Elasticity El 38.07 GPas.

2 Transverse Modulus of Elasticity Et 8.1 GPas

3 Modulus of Rigidity G12= G13= G23 3.05 GPas

4 Poisson Ratio ν12= ν13= ν23 0.22 -----

5 Density ρ 2200 2200 kg/m3

FINITE ELEMENT MODELS

The three layers laminated composite skew plates was simulated using a commercial finite element software

(ANSYS APDL version (17.2)). The element "SHELL281" was used to build the two dimensions skew plate with

three layers. The shape and properties of SHELL281 is illustrated in Figure (2) [40-42]. While the three

dimensions model was built using the element " SOLID186 " and the shape and properties of SOLID186 is also

illustrated in Figure (2) [40-42]. The number of elements in three dimensions model is about (275000- 300000)

elements and the number of nodes is about (550000 – 600000) nodes. The number of elements in the two

dimensions model is about (3700- 4000) elements and the number of nodes is about (11500 – 14000) nodes.

SHELL281 SOLID186



202

Element Geometry.

(b)Mesh Shape.

Figure 2. Mesh Information of the Two Elements.

VALIDATION

In the beginning, the non-dimensional fundamental frequency coefficient (Kf)) was calculated using the two finite

element models and then compared with those available in literature to check the validation for the present models.

The non-dimensional fundamental frequency coefficient (Kf)) is calculated as:

𝐾𝑓 = 𝜔𝑎2√𝜌1

𝐸𝑡 ∗ 𝑡3

Where: (𝜔) –frequency in (rad/sec), (a) – plate width in (m), (ρ1) -Mass density per unit area, t – plate thickness

in (m) and Et- Transverse Modulus of Elasticity in (N/m2).

A good agreement among the non-dimensional fundamental frequency coefficient results of the present models

and those of available in literature for the simply supported square anti-symmetric angle-ply laminates was shown

in Table 3.

Table 3. Fundamental Frequencies of Simply Supported Square Anti-Symmetric Angle-ply Laminates.

Fiber

Orientation

Jones

[43]

Reddy

[28]

Srinivasa et. al. [34] Present Work

CQUAD4 CQUAD8 SHELL281 SOLID186

0 18.805 18.806 18.685 18.804 18.3735 18.3715

15o 14.646 14.646 20.033 20.190 20.0073 20.0043

30o 14.204 14.204 23.137 23.353 22.8969 22.8919

45o 14.638 14.638 24.072 24.825 24.3505 24.3498

60o 14.204 14.204 23.137 23.353 22.8969 22.8919

75o 14.646 14.646 20.033 20.190 20.0073 20.0043

90o 18.805 18.806 18.685 18.804 18.3735 18.3715

RESULTS AND DISCUSSION

Five parameters, that effected on the fundamental frequency of skew composite plates, were investigated in this

work and these parameters are: skew angle, fiber orientation, layers layout, width-to-length Ratio (aspect ratio)



203

and type of support. Figure (3) illustrations the difference parameter (Kf) of the non-dimensional frequency due

to change in skew angle and (a/b) ratio for different supporting type and fiber orientation when the layer layout is

(0/ β /0). While Figures (4 and 5) show the same variation of non-dimensional frequency parameter (Kf) but when

the layers layout are (β/0/β) and (β/β/β) respectively. From Figures (3,4 and 5), the non-dimensional frequency

parameter (Kf) increases while angle of the skew increases when the other parameters are constant. When the

angle of skew increase, the boundary conditions effect will increase, and the affected area of plate will be reduced

too then the non-dimensional frequency parameter (Kf) will be increased. In other side, the effect of skew angle

increases when the (a/b) ratio increases. The non-dimensional frequency parameter (Kf), when (a/b) ratio is (2), is

greater than that when (a/b) ratio is (0.5) and (1.25) as shown in Figures (6, 7 & 8). This increasing in non-

dimensional frequency parameter (Kf) occurs because the width of the plate (a) was constant in this work and the

increasing in (a/b) ratio means decreasing in length of the plate (b) and this means reducing in area of the plate.

The reducing in the area of the plate leads to increasing in non-dimensional frequency parameter (Kf).

Generally, when skew angle, (a/b) ratio is constant for the same layers layouts and type of supporting, the fiber

orientation has a slight effect on the non-dimensional frequency parameter (Kf). In case (0/β/0), the fiber

orientation has a slight effect on the non-dimensional frequency parameter (Kf) because the change in mechanical

properties due to the fiber orientation is very small (see Figure (9)). While in cases (β /0/ β) and (β /β/ β), the

change in mechanical properties due to the fiber orientation is relatively large. Therefore, the fiber orientation has

different effect on the non-dimensional frequency parameter (Kf) in these cases depending on the supporting types

(see Figures (10 and 11)). Tis difference in variation of non-dimensional frequency parameter (Kf) due to fiber

orientation results from the combination of supporting type effect and fiber orientation effect as shown in Figures

(3,4 and 5). Generally, the mechanical properties of the case (0/β/0) is greater than that of the cases (β /0/ β) and

(β /β/ β) for any value of fiber orientation (β≠0) and when skew angle is zero. Therefore, the non-dimensional

frequency parameter (Kf) in case(0/β/0) is greater than that of the cases (β /0/ β) and (β /β/ β). When the skew

angle and fiber orientation change, the non-dimensional frequency parameter (Kf) sometime increases and

sometime decreases because of the combination of the skew angle effect and fiber orientation effect as shown in

Figures (9,10 and 11). Generally , the non-dimensional frequency parameter (Kf) of (SSSS) is greater than that of

(SFSF) and (FSFS) when skew angle is zero and (a/b) =1 because the effect of boundary conditions in (SSSS) is

greater than that of (SFSF) and (FSFS) . Also, the non-dimensional frequency parameter (Kf) of case (SFSF) is

smaller than that of (FSFS) because of the effect of fiber orientation (see Figures (3,4 and 5).

CONCLUSIONS

The skew angle, fiber orientation, (a/b) ratio, layers layout and supporting type are the parameters considered in

this work to study their effects on the fundamental frequency of the skew laminated composite plate. From the

outcomes, the following points can be concluded:

1- The first non-dimensional frequency parameters (Kf) increase when the skew angle increases at constant fiber

orientation, constant (a/b) ratio and for the same layers layouts and supporting types.

2- The first non-dimensional frequency parameters (Kf) increase when (a/b) ratio increases at constant fiber

orientation, constant the skew angle and for the same layers layouts and supporting types.

3- Generally, the fiber orientation has slight effect on the first non-dimensional frequency parameters (Kf). But

the effect of fiber orientation is strongly affected by the skew angle, layers layout and supporting type.

4- In spite of, the supporting type increases the value of non-dimensional frequency parameter (Kf) in cases of

(SSSS), but the supporting type has slight effect on the difference parameter of first non-dimensional

frequency. While the supporting type has strong effect on the difference of first non-dimensional frequency

parameter effect in cases (SFSF) and (FSFS).

5- The combination effects of five parameters leads to vary the first non-dimensional frequency parameters (Kf)

by different ways and different rate of variation.

6- The shell and solid models are very closed to each other and the shell model is simpler and faster than solid

model.



204

In future work, the different layers layout with number of layers more than three layers can be investigate to

investigate the effects of skew angle, fiber orientation , length-to-width ratio, types of supporting and layer layout

on the fundamental frequency of laminated skew composite plate.

SSSS SFSF FSFS

β =0 o.

β =30 o.

β =45 o.

β =60 o.

40.50

75.00

109.5144.0

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.000

40.50

75.00

109.5

144.0

178.5

213.0

247.5

282.0

Frequency Parameter (Kf)

8.444

10.74

13.03 15.33

17.62

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.6.150

8.444

10.74

13.03

15.33

17.62

19.91

22.21

24.50


3.888

7.075

10.26

13.4516.64

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

0.7000

3.888

7.075

10.26

13.45

16.64

19.82

23.01

26.20


40.63

75.25

109.9144.5

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.000

40.63

75.25

109.9

144.5

179.1

213.8

248.4

283.0


8.494

10.89

13.28 15.67

18.07

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.100

8.494

10.89

13.28

15.67

18.07

20.46

22.86

25.25


3.987

7.27510.56

13.85 17.14

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.0.7000

3.987

7.275

10.56

13.85

17.14

20.43

23.71

27.00


40.75

75.50

110.3145.0

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.000

40.75

75.50

110.3

145.0

179.8

214.5

249.3

284.0


8.537

10.98

13.41 15.85

18.29

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.100

8.537

10.98

13.41

15.85

18.29

20.73

23.16

25.60


4.025

7.35010.68

14.00 17.32

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

0.7000

4.025

7.350

10.68

14.00

17.32

20.65

23.98

27.30


41.25

76.50

111.8147.0

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.000

41.25

76.50

111.8

147.0

182.3

217.5

252.8

288.0


8.425

10.85

13.27 15.70

18.13

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.000

8.425

10.85

13.27

15.70

18.13

20.55

22.97

25.40


3.987

7.275

10.56

13.85

17.14

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Sk

ew A

ng

le.

0.7000

3.987

7.275

10.56

13.85

17.14

20.43

23.71

27.00




205

β =90 o.

Figure 3. The Variation of the First Non-Dimensional Frequency Parameter (Kf) Due to Variation in Skew

Angle and (a/b) Ratio for Different Fiber Orientation and Supporting Types When the Layer Layout is (0/ β /0).

SSSS SFSF FSFS

β =0 o.

β =30 o.

β =45 o.

41.75

77.50

113.3 149.0

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.000

41.75

77.50

113.3

149.0

184.8

220.5

256.3

292.0


8.375

10.70

13.02 15.35

17.67

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.050

8.375

10.70

13.02

15.35

17.67

20.00

22.32

24.65


3.912

7.125

10.34

13.55 16.76

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

0.7000

3.912

7.125

10.34

13.55

16.76

19.97

23.19

26.40


40.50

75.00

109.5144.0

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.000

40.50

75.00

109.5

144.0

178.5

213.0

247.5

282.0


8.444

10.74

13.03 15.33

17.62

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.150

8.444

10.74

13.03

15.33

17.62

19.91

22.21

24.50


3.888

7.075

10.26

13.4516.64

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

0.7000

3.888

7.075

10.26

13.45

16.64

19.82

23.01

26.20


39.13

73.25

107.4 141.5

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

5.000

39.13

73.25

107.4

141.5

175.6

209.8

243.9

278.0


7.313

10.82

14.34

17.8521.36

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

3.800

7.313

10.82

14.34

17.85

21.36

24.88

28.39

31.90


4.850

9.000

13.15

17.30 21.45

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

0.7000

4.850

9.000

13.15

17.30

21.45

25.60

29.75

33.90


39.13

73.25

107.4141.5

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Sk

ew A

ng

le.

5.000

39.13

73.25

107.4

141.5

175.6

209.8

243.9

278.0


7.425

11.55

15.68

19.8023.93

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

3.300

7.425

11.55

15.68

19.80

23.93

28.05

32.17

36.30


5.350

9.900

14.45

19.00 23.55

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Sk

ew A

ng

le.

0.8000

5.350

9.900

14.45

19.00

23.55

28.10

32.65

37.20




206

β =60 o.

β =90 o.

Figure 4. The Variation of the First Non-Dimensional Frequency Parameter Due to Variation in Skew Angle

and (a/b) Ratio for Different Fiber Orientation and Supporting Types When the Layer Layout is (β/0/β).

SSSS SFSF FSFS

β =0 o.

β =30 o.

37.13

70.25

103.4136.5

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

4.000

37.13

70.25

103.4

136.5

169.6

202.8

235.9

269.0


7.188

11.38

15.56

19.75 23.94

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

3.000

7.188

11.38

15.56

19.75

23.94

28.13

32.31

36.50


5.512

9.925

14.34

18.75 23.16

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

1.100

5.512

9.925

14.34

18.75

23.16

27.57

31.99

36.40


35.63

68.25 100.9

133.5

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

3.000

35.63

68.25

100.9

133.5

166.1

198.8

231.4

264.0


5.987

8.975

11.96

14.95

17.9420.92

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

3.000

5.987

8.975

11.96

14.95

17.94

20.92

23.91

26.90


4.900

8.300

11.70

15.10

18.50

18.50

21.90

21.90

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Sk

ew A

ng

le.

1.500

4.900

8.300

11.70

15.10

18.50

21.90

25.30

28.70


40.50

75.00

109.5144.0

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.000

40.50

75.00

109.5

144.0

178.5

213.0

247.5

282.0


8.444

10.74

13.03 15.33

17.62

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

6.150

8.444

10.74

13.03

15.33

17.62

19.91

22.21

24.50


3.888

7.075

10.26

13.4516.64

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Sk

ew A

ng

le.

0.7000

3.888

7.075

10.26

13.45

16.64

19.82

23.01

26.20


39.00

73.00

107.0 141.0

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

5.000

39.00

73.00

107.0

141.0

175.0

209.0

243.0

277.0


7.200

10.70

14.2017.70

21.20

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

3.700

7.200

10.70

14.20

17.70

21.20

24.70

28.20

31.70


4.825

8.950

13.07

17.2021.32

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Sk

ew A

ng

le.

0.7000

4.825

8.950

13.07

17.20

21.32

25.45

29.57

33.70




207

β =45 o.

β =60 o.

β =90 o.

Figure 5. The Variation of the First Non-Dimensional Frequency Parameter Due to Variation in Skew Angle

and (a/b) Ratio for Different Fiber Orientation and Supporting Types When the Layer Layout is (β/ β /β).

(a)a/b=0.5. (b) a/b=1.25.

38.63

72.25

105.9139.5

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

5.000

38.63

72.25

105.9

139.5

173.1

206.8

240.4

274.0


7.225

11.35

15.48

19.6023.73

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Sk

ew A

ng

le.

3.100

7.225

11.35

15.48

19.60

23.73

27.85

31.98

36.10


5.313

9.825

14.34

18.85

23.36

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Sk

ew A

ng

le.

0.8000

5.313

9.825

14.34

18.85

23.36

27.88

32.39

36.90


36.13

68.25

100.4132.5

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

4.000

36.13

68.25

100.4

132.5

164.6

196.8

228.9

261.0


7.025

11.25

15.48

19.70 23.93

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

2.800

7.025

11.25

15.48

19.70

23.93

28.15

32.38

36.60


5.500

9.900

14.30

18.70 23.10

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

1.100

5.500

9.900

14.30

18.70

23.10

27.50

31.90

36.30


34.13

65.25 96.38

127.5

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

3.000

34.13

65.25

96.38

127.5

158.6

189.8

220.9

252.0


5.813

8.825

11.84

14.85

17.8620.88

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Sk

ew A

ng

le.

2.800

5.813

8.825

11.84

14.85

17.86

20.88

23.89

26.90


4.900

8.300

11.70

15.10

18.50

18.50

21.90

21.90

0.50 0.75 1.00 1.25 1.50 1.75 2.00

0

15

30

45

60

75

a/b Ratio.

Skew

An

gle

.

1.500

4.900

8.300

11.70

15.10

18.50

21.90

25.30

28.70




208

(c) a/b=2.0.

Figure 6. Effect of Width-to-Length Ratio (a/b Ratio) on the First Non-Dimensional Frequency Parameter (Kf)

When the Fiber Orientation is (60 o) and Layer Layout is (0/ β/0).

(a)a/b=0.5. (b) a/b=1.25.

(c) a/b=2.0.

Figure 7. Effect of Width-to-Length Ratio (a/b Ratio) on the First Non-Dimensional Frequency Parameter (Kf)

When the Fiber Orientation is (60 o) and Layer Layout is (β/0/β).

(a)a/b=0.5. (b) a/b=1.25.



209

(c) a/b=2.0.

Figure 8. Effect of Width-to-Length Ratio (a/b Ratio) the First Non-Dimensional Frequency Parameter (Kf)

When the Fiber Orientation is (60 o) and Layer Layout is (β/ β /β).

(a)SSSS. (b)SFSF

(c)FSFS

Figure 9. Effect of Fiber Orientation on the First Non-Dimensional Frequency Parameter (Kf) When (a/b) ratio

is (1) and Layer Layout is (0/ β/0).

(a)SSSS. (b)SFSF



210

(c)FSFS


is (1) and Layer Layout is (β/0/β).

(a)SSSS. (b)SFSF

(c)FSFS


is (1) and Layer Layout is (β/ β /β).

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Studying the Effect of Fiber Orientation and Skew Angle on