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
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
Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using
Finite Element Method
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
Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using
Finite Element Method
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)
Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using
Finite Element Method
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.
Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using
Finite Element Method
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using
Finite Element Method
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using
Finite Element Method
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using
Finite Element Method
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
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
Frequency Parameter (Kf)
Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using
Finite Element Method
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.
Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using
Finite Element Method
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
Studying The Effect Of Fiber Orientation And Skew Angle On The Fundamental Frequencies Of Simply Supported Composite Plates Using
Finite Element Method
210
(c)FSFS
Figure 10. Effect of Fiber Orientation on the First Non-Dimensional Frequency Parameter (Kf) When (a/b) ratio
is (1) and Layer Layout is (β/0/β).
(a)SSSS. (b)SFSF
(c)FSFS
Figure 11. Effect of Fiber Orientation on the First Non-Dimensional Frequency Parameter (Kf) When (a/b) ratio
is (1) and Layer Layout is (β/ β /β).
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