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ISSN : 2093-5145(Print)
ISSN : 2288-0232(Online)
Journal of the Korean Society for Advanced Composite Structures Vol.10 No.3 pp.15-20

Free Vibration Characteristics of Laminated Composite Cylindrical Shells Reinforced by SWCNT with a Central Cutout

Ashish Maharjan1, Sang-Youl Lee2
1Master student, Department of Civil Engineering, Andong National University, Andong, Korea
2Associate Professor, Department of Civil Engineering, Andong National University, Andong, Korea

본 논문에 대한 토의를 2019년 07월 31일까지 학회로 보내주시면 2019년 8월호에 토론결과를 게재하겠습니다.

Corresponding author: Lee, Sang-Youl Department of Civil Engineering, Andong National University, 1375 Gyeongdong-Ro, Andong, Gyoungsangbuk-Do, 760-749, Korea. Tel: +82-54-820-5847, Fax: +82-54-820-6255, E-mail:
June 3, 2019 June 10, 2019 June 10, 2019


Single-walled Carbon Nanotubes (SWCNTs), epoxy resin and E-glass fibers were used for fabricating carbon nanotubes/fiber/polymer composite shells containing a central cutout. Modified Halpin-Tsai model and micro-mechanical approaches were used to evaluate the elastic properties of laminated composite cylindrical shells with different SWCNT weight ratios. The dynamic characteristics of shells were studied using the finite element program for the frequencies and mode shapes. The interactions between carbon nanotubes weight ratios, layup angles, cutout sizes, fundamental frequencies and mode shapes were carried out. The results were verified with those literatures which dealt with different curvature shapes with and without cutout. The study shows the importance of carbon nanotubes (CNT) reinforcement for higher frequency achievement.

중앙 개구부를 갖는 SWCNT로 보강된 적층 복합소재 원통형 쉘의 고유진동 특성

Ashish Maharjan1, 이 상열2
1안동대학교 토목공학과 석사과정
2안동대학교 토목공학과 부교수


본 연구에서는 중앙 개구부를 갖는 카본나노튜브/유리섬유/폴리머 합성 복합 적층쉘을 다루었다. 수정된 Halpin-Tsai 모델과 마이크로 역학적 접근방법은 단일벽 탄소나노튜브의 합성 비율에 따른 탄성적 물성변화를 추정하기 위하여 적용되었다. 유한요소 해석을 통하여 쉘의 고유진동 및 모드 특성을 분석하였다. 탄소나노튜브의 무게 비율, 보강섬유 각도, 개구부 크기, 고 유진동수 및 고유모드의 상관관계를 규명하였다. 개구부를 갖는 경우와 갖지 않는 경우에 대하여 곡률 변화에 따른 기존 문헌 과의 비교를 통하여 본 연구결과를 검증하였다. 본 연구결과는 고유진동 특성에 영향을 미치는 탄소나노튜브 보강의 중요성을 보여준다.

    National Research Foundation of Korea


    Combination of two or more materials with different properties resulting in a material with unique characteristics from the individual materials is a composite material. Basically, composite material is the combination of fiber/filament reinforcement and matrix, for examples, concrete, reinforced cement concrete, glass fiber reinforced plastic, carbon fiber reinforced polymer.

    Carbon nanotubes is rolled chicken wire-like structure of carbon atoms. The length is about 0.2-5 μm and diameter about 1-2nm. CNTs have the highest tensile strength and tensile modulus among the materials discovered till date (Lee, 2018). There are two types of CNT: Single-walled CNT (SWCNT) and Multi-walled CNT (MWCNT). SWCNT is a special type of carbon material made up of a single sheet of graphene rolled up and has a shape like a cylinder (Du et al., 2007;Kim et al., 2009). Cost of SWCNT is between 100-700 USD per gram. It is used in this study. MWCNT, on the other hand, is made up of single-walled carbon nanotubes arranged in concentric layers. It is stiffer and cheaper than SWCNT.

    Uniform dispersion of CNTs in the matrix is one of difficulties that is faced while working with composites involving CNT. The CNTs get entangled during the production phase and the van der Waals force between the CNTs attract each other resulting in aggregation. There should also be effective stress transfer from a matrix to a reinforcement. Achieving suitable CNT- matrix interfacial bonding for this transfer is another critical challenge (Lee, 2018). These challenges mentioned above hold back the performance of the resulting composites and the expected properties are not achieved.

    Dynamic characteristics of CNT reinforced laminated shell structures depend on the CNT weight ratios and the shape of the shell. Thus, this study focuses on the interaction between CNT weight ratios, curvatures and central cutout sizes in free vibrations of SWCNT reinforced laminated composite cylindrical shells.


    In this study, we combined epoxy resin and CNT first forming carbon nanotube reinforced composites (CNTRC). Later, CNTRC is combined with E-glass fibers forming carbon nanotubes/fiber/polymer composites (CNTFPC) (Han and Elliott, 2007;Zuo et al., 2013). The elastic properties of CNTRC and CNTFPC are calculated using the modified Halpin-Tsai equation and micro-mechanical approaches (Lee, 2018). From Halpin-Tsai equation, the Young's modulus of CNTRC:

    E c n r = E r e [ 3 8 ( 1 + 2 ( l c n t / d c n t ) γ d t V c n t 1 γ d l V c n t ) + 5 8 ( 1 + 2 γ d d V c n t 1 γ d d V c n t ) ]


    γ d d = ( E 11 c n t / E r e ) ( d c n t / 4 t c n t ) ( E 11 c n t / E r e ) + ( d c n t / 2 t c n t ) γ d l = ( E 11 c n t / E r e ) ( d c n t / 4 t c n t ) ( E 11 c n t / E r e ) + ( l c n t / 2 t c n t )

    where, Ecep, Ecp and E 11 c n are the Young’s moduli of CNTRC, epoxy resin and CNT, and lcn, dcn and tcn are the length, diameter and thickness of the CNT respectively.

    The longitudinal Young’s modulus of CNTFPC can be calculated by

    E 11 = E f V f + E c n r ( 1 V f )

    E22, G12, ν12, using Halpin-Tsai model, can be determined as

    Φ f ( E 22 , E 12 , ν 12 ) Φ f ( E c e p , G c e p , ν c e p ) = 1 + χ η V f 1 η V f

    η = Φ f ( E 22 , E 12 , ν 12 ) / Φ c e p ( E c e p , G c e p , ν c e p ) 1 Φ f ( E 22 , G 12 , ν 12 ) / Φ c e p ( E c e p , G c e p , ν c e p ) + χ

    where, Φ, Φcep, and Φf denote material modulus of composites, corresponding matrix modulus, and corresponding fiber modulus, respectively. χ in Eq.(4) and Eq.(5) is called the reinforcing factor and it depends on the fiber geometry and packing geometry. The value of χ lies between 1.0 and 2.0. For circular fibers of square array, as in this study, the value of χ is 2. From Eq.(4) and Eq.(5), for E22, the equations can be written as:

    E 22 = E c e p 1 + χ η V f 1 η V f , η = E f E c e p E f χ E c e p

    In the case of shear modulus (G12), if the volume fraction of fiber (Vf) in the CNTFPC is more than 0.5, the Halpin- Tsai equation gives the result lower than the actual. So, in this case, the equation from Hewitt and Malherbe (1970) is used to calculate χ :

    χ = 1 + 40 ( V f ) 10

    This equation is derived from the experimental results.

    Table 1 shows properties and definition of the materials used in this study.


    The study has been carried out for the fundamental frequencies of vibration. The modeling of different cases of the composite shell structures for the frequencies was done in the ABAQUS program. As shown in Table 2, natural frequencies are computed from the program and the results are compared with those of previous studies. The table shows sufficient accuracy of the procedure followed in the ABAQUS program.

    The effect of different shapes, layup angles, sizes of cutout, CNT weight ratios were considered for the study. Square shell structure with sides 1.0m and thickness 10mm of different shapes with radius of 0.8m, 0.55m, 0.4m, 0.32m including flat are shown in Fig. 1. Fig. 2 shows shells with cutout reducing the mass by 1%, 4%, 16% and 36%. Layup angles of [0/90], [0/90/90/0], [0/90/0/90], [45/-45], [45/-45/45/-45] and [45/-45/ -45/45] are used for the numerical analysis.

    Table 3 shows the material properties of CNTFPC for different CNT weight ratios calculated using the multiscale analysis discussed earlier. Fig. 3 shows the variation of fundamental frequency due to different curvatures. The flat model has the least frequency and as the curvature increases, the frequency also increases with more curvature. The figure also shows weight ratio. For the radius of 0.32m, the percentage that increas with it. This happens because the membrane force of frequency with the increase in CNT increase in frequencies from 0% through 8% CNT weight ratio is 6.38%, 2.95%, 1.82% 1.28%, 0.95%, 0.81%, 0.69% and 0.57% respectively. The increase in frequency is high at first but decreases with the increase in CNT weight ratio. As CNT is very expensive, adding more CNT is clearly not beneficial. Addition of less than 2% CNT would be better considering the costing. Effect of different layup angles on the frequency in the same thickness of composite is shown in Fig. 4. Comparing six different layup angles, [0/90/90/0] gives the highest frequency.

    Cutouts are structural requirements in every structure. It may be provided to reduce weight, access the interior or to lay the lines for fuel or electricity. There are changes in the frequencies with the introduction of cutout as shown in Table 4. From the fundamental formula for the natural frequency, both the stiffness and mass change simultaneously when cutout is introduced to the structure which may result in increase or decrease of the frequency (Sahu and Datta, 2002;Rao and Krishnan, 1999). In this case, the frequency decreases with the increase in the size of the cutout. For 4.0% reduction cutout in the structure, the frequency with no CNT is 9.5924Hz which is less than the one without cutout. Simply, with the addition of 1% CNT, we get 10.195 Hz which means the frequency loss due to cutout is recovered (Lee and Park, 2009). Similarly, for 16% cutout, 2.0% CNT can be added to get 9.7978Hz. This shows that CNT reinforcement can be used to recover the frequency loss from cutout of laminated composite plates.

    Fig. 5 and 6 are the mode shapes with and without SWCNT addition. There is no significant change in the mode shapes. But there are changes with the change in cutout size.


    The result of the study can be summarized as follows:

    • 1) CNT reinforcement in laminated composite shell structure increases the fundamental frequency.

    • 2) As the curvature increases, the membrane force increases with it resulting in higher frequency.

    • 3) CNT is expensive and use of less than 2% CNT would be better. [0/90/90/0] layup angle gives the highest frequency among the six different layup cases.

    • 4) CNT reinforcement can be used to recover the frequency loss from cutout of laminated composite plates or shells.

    • 5) There is negligible change in the mode shape due to the addition of CNT but significant change is seen when the cutout size changes.


    본 연구는 한국연구재단 기초연구사업의 지원을 받아 수행된 연구 (No.2018R1D1A1B07050080)이며, 행정안 전부 장관의 방재안전분야 전문인력양성 사업으로 지 원되었습니다.


    Geometries and Boundary Condition of Cylindrical Shells with Varying Radius
    Shells with a Central Cutout
    Induced Natural Frequencies(Hz) for Different SWCNT Weight Ratios and Curvatures (0/90)
    Induced Natural Frequencies(Hz) for Different SWCNT Weight Ratios and Layup Sequences
    Mode Shapes of Flat Plates without CNT for Different Cutout Sizes
    Mode Shapes of Flat Plates with 1% CNT Weight Ratio for Different Cutout Sizes


    Mechanical Properties of the Materials Used in this Study
    Comparison of Frequencies for the Clamped Cylindrical Curved Panel with and without Cutout (No CNT reinforcement, a=b=500mm, h=2mm, ν=0.3, E=7,020kg/mm2, ρ=2,720kg/m3)
    Material Properties of CNTFPC for Different CNT Weight Ratios
    Natural Frequencies with Cutout and without Cutout for Different CNT Weight Ratios of a Flat Shape Model


    1. Du, J. H. , Bai, J. , and Cheng, H. M. (2007), “The present status and key problems of carbon nanotube-based polymer composites”, Express Polymer Letters, Vol. 1, pp. 253-273.
    2. Han, Y. and Elliott, J. (2007), “Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites”, Computational Materials Science, Vol. 39, No. 2, pp. 315-323.
    3. Hewitt, R. L. and Malherbe, M. C. (1970), “An approxi- mation for the longitudinal shear modulus of continuous fiber composites”, Journal of Composite Materials, Vol. 4, pp. 280-282.
    4. Kim, M. , Park, Y. B. , Okoli, O. I. , and Zhang, C. (2009), “Processing, characterization, and modeling of carbon nanotube-reinforced multiscale composites”, Composites Science Technology, Vol. 69, pp. 335-342.
    5. Lee, S. Y. (2018), “Dynamic instability assessment of carbon nanotube/fiber/polymer multiscale composite skew plates with delamination based on HSDT”, Composite Structures. 200, pp. 757-770.
    6. Lee, S. Y. and Park, T. (2009), “Free vibration of laminated composite skew plates with central cutouts”, Structural Engineering and Mechanics, Vol. 31, pp. 587-603.
    7. Lee, S. Y. (2018), “Determination of material properties and stiffness variations of carbon nanotube reinforced multi-scale laminated composite structures”, J. of Korean Society for Advanced Composite Structures. Vol. 9, pp. 14-19.
    8. Sahu, S. K. and Datta, P. K. (2002), “Dynamic stability of curved panels with cutouts”, Journal of Sound and Vibration, Vol. 251, pp. 683-696.
    9. Sivasubramonian, B. , Rao, G. V. , and Krishnan, A. (1999), “Free vibration of longitudinally stiffened curved panels with cutouts”, Journal of Sound and Vibration, Vol. 226, pp. 41-55.
    10. Zuo, Z. H. , Huang, X. , Rong, J. H. , and Xie, Y. M. (2013), “Multiscale design of composite materials and structures for maximum natural frequencies”, Materials and Design, Vol. 51, pp. 1023-1034.