free vibration of skew isotropic plate using ansys

Journal of Mechanical Engineering Research and Developments

ISSN: 1024-1752

CODEN: JERDFO

Vol. 43, No.5, pp. 472-486 Published Year 2020

472

Free Vibration of Skew Isotropic Plate Using ANSYS

Faeq H. Gburi†*, Luay S. Al-Ansari‡, Mohannad A. Khadom†, Ali A. Al-Saffar‡

†AL-Dewaniyah Technical Institute, AL-Furat Al-Awsat Technical University Iraq

University of Kufa, Iraq –gineering Faculty of En –Mechanical Engineering Department ‡

[email protected]: -E *Corresponding author

ABSTRACT: Free vibration analysis of the isotropic skew plate with three supporting conditions (CCCC, CFCF

and FCFC) were studied. ANSYS APDL version 17.2. with two-dimensional model (SHELL281 element) and

three-dimensional model (SOLID186 element) were adopted. The effects of width-thickness ratio (a/t), aspect

ratio (a/b), mode number, skew angle and supporting conditions on the first five natural frequencies were

investigated. The results showed that the non-dimensional fundamental frequency coefficients (Kf) increases

when the skew angle at any aspect ratio (a/b) , mode number and supporting type. Also, the effect of skew angle

appears sharply when the skew edges are clamped (i.e. CCCC and CFCF) while in when the skew edge is free ,

the effect of skew angle is not appear . Finally, the comparison between the two elements (SHELL281 and

SOLID186) shows that there is a good agreement between them when the skew angle is smaller than 60o and the

mode number is smaller than fourth mode.

KEYWORDS: Free Vibration, Skew plate, ANSYS APDL, Non-Dimensional Frequency Coefficient.

INTRODUCTION

The skew plates have wide range of applications in civil, marine, aeronautical and mechanical engineering. It can

be found in alignment problems in design of bridges, swept wings of aero-planes, parallelogram slabs in

constructions and ship hulls. Therefore, several methods were used to analyze the vibration of skew plate. Most

of the literature dealt with the vibration of skew plate are based on inexact methods. Vibration characteristics of

skew plates were studied by several researches using approximate or numerical methods like FEM, Rayleigh-Ritz

method, series method, etc. The earlier studies on free vibration characteristics of skew plates were obtained by

Barton [1], Kaul and Cadambe [2] and Hasegawa [3], using Rayleigh-Ritz method. The Lagrangian-multiplier

method was utilized by Hamada [4] to calculate the Natural frequency of the rhombic skew plate. Claassen [5]

adopted a Fourier sine series solution scheme in conjunction with the Rayleigh-Ritz method to extend the work

of Barton [1].

Conway and Farnham [6] considered the free vibration of three shapes of plates (triangular, rhombic and

parallelogram plates) by employing the point matching method. They calculated the frequencies of skew plate

types for different skew angles of clamped and simply supported boundary conditions. Laura and Grosson [7]

applied conformal mapping and Galerkin’s method to calculate natural frequencies for vibration of simply

supported rhombic plates. They compared their results with Conway and Farnham [6] and they found that the

difference between the two results increased with the increasing of the angle skew plate. Monforton [8] employed

FEM to obtain natural frequencies of clamped rhombic plates. Durvasula [9] adopted Galerkin’s method to find

the frequencies and mode shapes of clamped skew plates. He expressed the deflection function as a double series

of beam characteristic functions in terms of skew coordinates to satisfy zero deflection and normal slope on all

the edges. He also compared his results with the results of Kaul and Cadambe [2], Hasegawa [3] and Conway and

Farnham [6].

Frequencies of orthotropic and isotropic of skew plates were reported by Nair and Durvasula [10]. Simply

supported, clamped and free edge supporting conditions were adopted as well as the combination of these three

boundary conditions. Srinivasan and Ramachandran [11] studied variations of frequencies and mode shapes of

orthotropic skew plates utilizing a numerical method. Kuttler and Sigillito [12] solved vibration problem of skew

plates using trial function method. Mizusuwa et al. (13,14), Mizusuwa and Kajita (15,16) employed the B-spline

functions and Rayleigh-Ritz method of isotropic skew plates to investigate the effect of skew angle and location


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on natural frequencies. Bardell [17] calculated natural frequencies and mode shapes of isotropic skew plates

applying the hierarchical finite element method. Liew and Wang [18] calculated the frequency of rhombic plates

using the Rayleigh-Ritz method. They examined the effect of internal support, skew angle and aspect ratio on the

frequency of four rhombic plates with different support conditions. Singh and Chakraverthy [19] implemented

boundary characteristics orthogonal polynomials to find the first five frequencies for the transverse vibration of

skew plates under different boundary conditions. Garg et.al. [20] analyzed the free vibration of isotropic,

orthotropic, and layered anisotropic composite and sandwich skew laminates using iso-parametric finite element

model. Malhotra et al. [21] employed finite element to determine frequency of the rhombic and

parallelogrammical orthotropic plates for various boundary conditions and skew angles. The orthotropic,

anisotropic and laminated skew plates were studied from several researchers at different types of supporting

conditions using different methods [22-32]. In this work, the first five natural frequencies of the skew isotropic

plates with three types of supporting (CCCC, CFCF and FCFC) were investigated applying Finite Element Method

(FEM). ANSYS APDL (version 17.2) was adopted to build two models and these models were two-dimensional

shell model and three-dimensional solid model of skew plate. These two models were used to study the effects

of aspect ratio (a/b), skew angle width-thickness ratio and mode number on the first five natural frequencies.

PROBLEM DESCRIPTION

The skew plate is one of the member used in several types of structure. The natural frequency of this type of plates

are affected by different parameters such as skew angle, aspect ratio, supporting type, …etc. The geometry and

dimensions of skew aluminum plate used in this work are illustrated in Figure.1. The mechanical properties of the

skew plate are: Modulus of Elasticity (E)= 71.7*109 N/m2, Poison ratio (ν)=0.33 and Density (ρ) = 2800 kg/m3.

The width of skew plate (a) is (0.3 m) and the aspect ratio of the plate (a/b) is (0.5,0.75, 1.0, 1.25, 1.5, 1.75 and

2). The skew angle (θ) was changed from (0o) to (75o) with step (15o). Three types of boundary conditions were

employed and these conditions are CCCC, CFCF and FCFC starting from left side of the plate.

Figure 1. Dimension of Skew Plate.

Finite Element Models (ANSYS Models)

In this work, two finite element models were built using ANSYS APDL version (17.2). The first model is two-

dimensional model and the element (SHELL281) was used in this model (see Figure 2). "SHELL281 is well-

suited for linear, large rotation, and/or large strain nonlinear applications. Change in shell thickness is accounted

for in nonlinear analyses. The element accounts for follower (load stiffness) effects of distributed pressures" [33].

The number of elements in the first model are about (3700- 4000) elements and the number of nodes are about

(11500–14000) nodes. The second model is three-dimensional model and the element (SOLID186) as shown in

Figure.2 was used. "SOLID186 is a higher order 3-D 20-node solid element that exhibits quadratic displacement

behavior. The element is defined by 20 nodes having three degrees of freedom per node: translations in the nodal

x, y, and z directions. The element supports plasticity, hyper elasticity, creep, stress stiffening, large deflection,

and large strain capabilities. It also has mixed formulation capability for simulating deformations of nearly

incompressible elastoplastic materials, and fully incompressible hyper elastic materials" [33]. The number of


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elements in the second model are about (275000-300000) elements and the number of nodes are about (550000 –

600000) nodes.

SHELL281 SOLID186

(a) Element Geometry

(b) Meshing of Skew Plate.

(c) Mesh Shape.

Figure 2. Dimension of Skew Plate.

Validation

In order to check the validation for the present two models, the non-dimensional fundamental frequency

coefficients (Kf) were compared with those available from literature. The non-dimensional fundamental frequency

coefficient (Kf) was calculated as:

𝐾𝑓 =𝜔𝑎2

𝜋2√

𝜌𝑡

𝐷


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Where:

a – The width of plate (m),

t – The thickness of plate (m),

𝜔 – The frequency (rad/sec),

𝐷 =𝐸𝑡3

12(1 − 𝜈2)

E- Modulus of Elasticity (N/m2).

ν- Poisson ratio.

Table 1 shows a good agreement between the first non-dimensional fundamental frequency coefficient results of

the present models and those of available from literature for the isotropic clamped skew plate.

Table 1: The Non-Dimensional Fundamental Frequency Coefficient (Kf) of Isotropic Clamped (C-C-C-C) Skew

Plates.

Authors Skew Angle.

0o 15 o 30 o 45 o

Durvasula [9] 3.6467 3.87 4.675 6.68

Liew &Lam [34] 3.636 3.8691 4.6698 6.6519

Reedy and Palaninathan [26] 3.638 3.872 4.672 6.663

A. K. Garg et al.[20] 3.648 3.872 4.68 6.699

Srinivasa et. al[30] -CQUAD4 3.619 3.8395 4.6301 6.5798

Srinivasa et. al[30] -CQUAD8 3.642 3.8645 4.6632 6.6378

Present Work - SHELL281 3.6441 3.867 4.6681 6.65036

Present Work - SOLID186 3.6778 3.8676 4.6685 6.6511

RESULTS AND DISCUSSION

The results of this work were divided into three parts. The first part dealt with the effect of the width–thickness

ratio (a/t) on the non-dimensional fundamental frequency Coefficient (Kf). The second part described a

comparison between the non-dimensional fundamental frequency Coefficient (Kf) values that was calculated by

using elements SHELL281 and SOLID186. The third part described the effect of aspect ratio (a/b), skew angle

and mode number on the non-dimensional fundamental frequency Coefficient (Kf). These results are discussed

below:

Effect of the Width – Thickness Ratio

Figure.3 shows the effect of width–thickness ratio (a/t) on the first six non-dimensional fundamental frequency

Coefficients (Kf) of the CCCC square skew plate. It can be seen that there is no variation in the non-dimensional

fundamental frequency Coefficients (Kf) for the high width-thickness ratio. Same results are shown in Figure.4

and Figure.5 when the supporting conditions are CFCF and FCFC respectively. For low width-thickness ratio, the

variation in variation in the non-dimensional fundamental frequency Coefficients (Kf) increases when the skew

angle increases. Also, when the number of mode increases, the variation in variation in the non-dimensional

fundamental frequency Coefficients (Kf) increases for low width-thickness ratio. From Figures. 3,4 and 5, the

variation of non-dimensional fundamental frequency Coefficient (Kf) due to variation in width-thickness ratio of

CCCC was greater than that of FCFC and CFCF. In other word, the effect of width-thickness ratio on the first six

non-dimensional fundamental frequency Coefficients (Kf) appears at low width-thickness ratio, high skew angle,

high mode number and CCCC supporting.

Effects of aspect ratio (a/b), skew angle and mode number


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Figure.6 shows the effect of aspect ratio (a/b) for CCCC skew plate on the first five non-dimensional fundamental

frequency Coefficients (Kf) with different skew angle. From the results, the frequency Coefficient (Kf) increases

when the aspect ratio (a/b) increased for any skew angle and any mode number. Similar behavior is shown in

Figure.7 and Figure.8 for the supporting conditions CFCF and FCFC respectively. From figures (6-8), the non-

dimensional fundamental frequency Coefficient (Kf) increases with increasing the mode number for any skew

angle and aspect ratio (a/b). Finally, Figures.9-11 show the effect of skew angle on the first five non-dimensional

fundamental frequency Coefficients (Kf) for CCCC, CFCF and FCFC skew plate at different aspect ratio (a/b).

From these figures, the following points can be noticed:

(a) The effect of skew angle on the non-dimensional fundamental frequency Coefficient (Kf) in CFCF is

smaller than CCCC and FCFC for any mode number because the supporting edge length in case CFCF

is smaller than CCCC and FCFC.

(b) The effect of skew angle on the non-dimensional fundamental frequency Coefficient (Kf) increases

rapidly when the skew angle larger than 45o because the effect of supporting edge length increases when

the skew angle increases.

(c) The effect of skew angle on the non-dimensional fundamental frequency Coefficient (Kf) increases when

the mode number increased.

(d) The effect of skew angle on the non-dimensional fundamental frequency Coefficient (Kf) increases when

the aspect ratio (a/b) increased.

Comparison between the results of the two- and three-dimensional models.

In order to discover the convergence between SHELL281 and SOLID186, comparison between the first five non-

dimensional fundamental frequency Coefficients (Kf) was done for CCCC, CFCF and FCFC square skew plate as

listed in Table (2). In the first mode, there is a good agreement between the results of the two models when the

skew angle is smaller than 60o for all types of supporting conditions and after 60o, the difference between the non-

dimensional fundamental frequency Coefficient (Kf) of the two models increases. The same behavior can be

noticed in the results of the second, third, fourth and fifth mode. Also, when the number of mode increases, the

difference between the results of the two element increases.

CONCLUSIONS

From the results, the following points can be concluded:

1- At high width–thickness ratio (a/t) , the width–thickness ratio (a/t) does not affect the non-dimensional

fundamental frequency Coefficient (Kf) of the CCCC, CFCF and FCFC skew plates. While, the non-

dimensional fundamental frequency Coefficient (Kf) increases when the width–thickness ratio (a/t)

increases.

2- The non-dimensional fundamental frequency Coefficient (Kf) increases when the aspect ratio (a/b)

increases for any skew angle, mode number and type of supporting.

3- The non-dimensional fundamental frequency Coefficient (Kf) increases when the mode number increases

for any skew angle, aspect ratio (a/b) and type of supporting.

4- The non-dimensional fundamental frequency Coefficient (Kf) increases when the skew angle increases

for any mode number, aspect ratio (a/b) and type of supporting.

5- The effect of skew angle on the non-dimensional fundamental frequency Coefficient (Kf) is small for

CFCF in comparison with CCCC and FCFC for any mode number and mode number. In other word , the

effect of skew angle appears sharply when the skew edges are clamped (i.e. CCCC and CFCF) while in

when the skew edge is free , the effect of skew angle is not appear.

6- The effect of skew angle on the non-dimensional fundamental frequency Coefficient (Kf) is greater than

that of aspect ratio , supporting type and number of mode.


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7- There is an excellent agreement between the results of the two-dimensional model (element SHELL281)

and three-dimensional model (element SOLID186) for CCCC, CFCF and FCFC skew plates when the

skew angle and mode number are small.

Finally, the effect of cutout or crack on the fundamental frequencies of isotropic skew plate can studied using the

same models. Also, the free vibration problem of triangular and trapezoidal plates can be studied using the same

finite element models.

θ=0 o. θ=30 o.

θ=15 o. θ=45 o.

θ=60 o. θ=75 o.

Figure 3. Effect of Width – Thickness Ratio on the Non-Dimensional Fundamental Frequency Coefficient (Kf)

for CCCC Square Skew Plates.


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θ=0 o. θ=15 o.

θ=30 o. θ=45 o.

θ=60 o. θ=75 o.


for CFCF Square Skew Plates.

θ=0 o. θ=15 o.


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θ=30 o. θ=45 o.

θ=60 o. θ=75 o.


for FCFC Square Skew Plates.

First Mode. Second Mode.

Third Mode. Fourth Mode.


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Fifth Mode.

Figure 6. Effect of aspect ratio (a/b) on the Non-Dimensional Fundamental Frequency Coefficient (Kf) for

CCCC Skew Plates.



Fifth Mode.


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CFCF Skew Plates.



Fifth Mode.


FCFC Skew Plates.



482


Fifth Mode.

Figure 9. Effect of Skew Angle on the Non-Dimensional Fundamental Frequency Coefficient (Kf) for CCCC

Skew Plates.




483

Fifth Mode.

Figure 10. Effect of Skew Angle on the Non-Dimensional Fundamental Frequency Coefficient (Kf) for CFCF

Skew Plates.



Fifth Mode.


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Figure 11. Effect of Skew Angle on the Non-Dimensional Fundamental Frequency Coefficient (Kf) for FCFC

Skew Plates.

Table 2. The Non-Dimensional Fundamental Frequency Coefficient (Kf) for Square Skew Plates Using

SHELL281 and SOLID186.

(a) First Mode.

SHELL281 SOLID186

Skew Angle CCCC CFCF FCFC CCCC CFCF FCFC

0 3.644096 2.239301 2.239301 3.677812 2.266466 2.266466

15 3.867193 2.3574 2.3574 3.867563 2.357511 2.360657

30 4.668093 2.763402 2.763402 4.668464 2.762995 2.763106

45 6.650358 3.666154 3.666154 6.651099 3.663896 3.66397

60 12.33031 5.724734 5.724734 12.3329 5.70956 5.70956

75 41.41809 13.72522 13.72818 43.31301 14.13604 14.13604

(b) Second Mode.

SHELL281 SOLID186


0 7.431643 2.652852 2.652852 7.488269 2.681387 2.681387

15 7.381309 2.745119 2.745119 7.38205 2.745415 2.749042

30 8.263262 3.067181 3.067181 8.264372 3.066774 3.066885

45 10.78662 3.814269 3.814269 10.78847 3.812048 3.812048

60 18.00841 5.789502 5.789502 18.01396 5.776178 5.776178

75 52.84683 14.01538 14.01649 56.12223 14.4125 14.4125

(c) Third Mode.

SHELL281 SOLID186


0 7.431643 4.382745 4.382745 7.512696 4.411613 4.411613

15 8.36578 4.50858 4.50858 8.367261 4.50969 4.513021

30 10.65042 4.98379 4.98379 10.6519 4.98453 4.98453

45 15.0191 6.225112 6.225112 15.02243 6.225482 6.225482

60 23.47629 9.764765 9.764765 23.48518 9.755142 9.755142

75 62.86548 23.06659 23.06289 68.17644 23.91709 23.91709

(d) Fourth Mode.

SHELL281 SOLID186


0 10.95649 6.179959 6.179959 11.05827 6.25583 6.25583

15 11.09269 6.51083 6.51083 11.09454 6.51157 6.520453

30 12.07457 7.442006 7.442006 12.07642 7.441636 7.442006

45 15.92659 8.721818 8.721818 15.93029 8.719597 8.719597

60 29.52376 11.63637 11.63637 29.53708 11.61934 11.61934

75 73.32456 23.19613 23.19798 80.96346 24.30384 24.30384

(e) Fifth Mode.

SHELL281 SOLID186


0 13.32144 6.76435 6.76435 13.43395 6.842441 6.842441

15 14.07016 7.080417 7.080417 14.07312 7.081527 7.09152

30 16.70269 8.128175 8.128175 16.70602 8.127435 8.127805

45 19.92332 10.25182 10.25182 19.9285 10.24738 10.24738

60 30.87981 14.58275 14.58275 30.89202 14.57387 14.57387

75 83.83915 32.19256 32.17295 94.52399 33.75883 33.75883


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free vibration of skew isotropic plate using ansys

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