A higher-order theory for static and dynamic analyses of functionally graded beams

Xian Fang Li, Baolin Wang, Jiecai Han

    Research output: Contribution to journalArticleResearchpeer-review

    Abstract

    The higher-order theory is extended to functionally graded beams (FGBs) with continuously varying material properties. For FGBs with shear deformation taken into account, a single governing equation for an auxiliary function F is derived from the basic equations of elasticity. It can be used to deal with forced and free vibrations as well as static behaviors of FGBs. A general solution is constructed, and all physical quantities including transverse deflection, longitudinal warping, bending moment, shear force, and internal stresses can be represented in terms of the derivatives of F. The static solution can be determined for different end conditions. Explicit expressions for cantilever, simply supported, and clamped-clamped FGBs for typical loading cases are given. A comparison of the present static solution with existing elasticity solutions indicates that the method is simple and efficient. Moreover, the gradient variation of Young’s modulus and Poisson’s ratio may be arbitrary functions of the thickness direction. Functionally graded Rayleigh and Euler–Bernoulli beams are two special cases when the shear modulus is sufficiently high. Moreover, the classical Levinson beam theory is recovered from the present theory when the material constants are unchanged. Numerical computations are performed for a functionally graded cantilever beam with a gradient index obeying power law and the results are displayed graphically to show the effects of the gradient index on the deflection and stress distribution, indicating that both stresses and deflection are sensitive to the gradient variation of material properties.
    Original languageEnglish
    Pages (from-to)1197-1212
    Number of pages16
    JournalArchive of Applied Mechanics
    Volume80
    Issue number10
    Early online date8 May 2010
    DOIs
    Publication statusPublished - Oct 2010

    Fingerprint

    Elasticity
    Materials properties
    Elastic moduli
    Cantilever beams
    Poisson ratio
    Bending moments
    Shear deformation
    Stress concentration
    Residual stresses
    Derivatives

    Cite this

    Li, Xian Fang ; Wang, Baolin ; Han, Jiecai. / A higher-order theory for static and dynamic analyses of functionally graded beams. In: Archive of Applied Mechanics. 2010 ; Vol. 80, No. 10. pp. 1197-1212.
    @article{a1a496821a944e4a8007659d46dd5c38,
    title = "A higher-order theory for static and dynamic analyses of functionally graded beams",
    abstract = "The higher-order theory is extended to functionally graded beams (FGBs) with continuously varying material properties. For FGBs with shear deformation taken into account, a single governing equation for an auxiliary function F is derived from the basic equations of elasticity. It can be used to deal with forced and free vibrations as well as static behaviors of FGBs. A general solution is constructed, and all physical quantities including transverse deflection, longitudinal warping, bending moment, shear force, and internal stresses can be represented in terms of the derivatives of F. The static solution can be determined for different end conditions. Explicit expressions for cantilever, simply supported, and clamped-clamped FGBs for typical loading cases are given. A comparison of the present static solution with existing elasticity solutions indicates that the method is simple and efficient. Moreover, the gradient variation of Young’s modulus and Poisson’s ratio may be arbitrary functions of the thickness direction. Functionally graded Rayleigh and Euler–Bernoulli beams are two special cases when the shear modulus is sufficiently high. Moreover, the classical Levinson beam theory is recovered from the present theory when the material constants are unchanged. Numerical computations are performed for a functionally graded cantilever beam with a gradient index obeying power law and the results are displayed graphically to show the effects of the gradient index on the deflection and stress distribution, indicating that both stresses and deflection are sensitive to the gradient variation of material properties.",
    author = "Li, {Xian Fang} and Baolin Wang and Jiecai Han",
    year = "2010",
    month = "10",
    doi = "10.1007/s00419-010-0435-6",
    language = "English",
    volume = "80",
    pages = "1197--1212",
    journal = "Archive of Applied Mechanics",
    issn = "0939-1533",
    publisher = "Springer",
    number = "10",

    }

    A higher-order theory for static and dynamic analyses of functionally graded beams. / Li, Xian Fang; Wang, Baolin; Han, Jiecai.

    In: Archive of Applied Mechanics, Vol. 80, No. 10, 10.2010, p. 1197-1212.

    Research output: Contribution to journalArticleResearchpeer-review

    TY - JOUR

    T1 - A higher-order theory for static and dynamic analyses of functionally graded beams

    AU - Li, Xian Fang

    AU - Wang, Baolin

    AU - Han, Jiecai

    PY - 2010/10

    Y1 - 2010/10

    N2 - The higher-order theory is extended to functionally graded beams (FGBs) with continuously varying material properties. For FGBs with shear deformation taken into account, a single governing equation for an auxiliary function F is derived from the basic equations of elasticity. It can be used to deal with forced and free vibrations as well as static behaviors of FGBs. A general solution is constructed, and all physical quantities including transverse deflection, longitudinal warping, bending moment, shear force, and internal stresses can be represented in terms of the derivatives of F. The static solution can be determined for different end conditions. Explicit expressions for cantilever, simply supported, and clamped-clamped FGBs for typical loading cases are given. A comparison of the present static solution with existing elasticity solutions indicates that the method is simple and efficient. Moreover, the gradient variation of Young’s modulus and Poisson’s ratio may be arbitrary functions of the thickness direction. Functionally graded Rayleigh and Euler–Bernoulli beams are two special cases when the shear modulus is sufficiently high. Moreover, the classical Levinson beam theory is recovered from the present theory when the material constants are unchanged. Numerical computations are performed for a functionally graded cantilever beam with a gradient index obeying power law and the results are displayed graphically to show the effects of the gradient index on the deflection and stress distribution, indicating that both stresses and deflection are sensitive to the gradient variation of material properties.

    AB - The higher-order theory is extended to functionally graded beams (FGBs) with continuously varying material properties. For FGBs with shear deformation taken into account, a single governing equation for an auxiliary function F is derived from the basic equations of elasticity. It can be used to deal with forced and free vibrations as well as static behaviors of FGBs. A general solution is constructed, and all physical quantities including transverse deflection, longitudinal warping, bending moment, shear force, and internal stresses can be represented in terms of the derivatives of F. The static solution can be determined for different end conditions. Explicit expressions for cantilever, simply supported, and clamped-clamped FGBs for typical loading cases are given. A comparison of the present static solution with existing elasticity solutions indicates that the method is simple and efficient. Moreover, the gradient variation of Young’s modulus and Poisson’s ratio may be arbitrary functions of the thickness direction. Functionally graded Rayleigh and Euler–Bernoulli beams are two special cases when the shear modulus is sufficiently high. Moreover, the classical Levinson beam theory is recovered from the present theory when the material constants are unchanged. Numerical computations are performed for a functionally graded cantilever beam with a gradient index obeying power law and the results are displayed graphically to show the effects of the gradient index on the deflection and stress distribution, indicating that both stresses and deflection are sensitive to the gradient variation of material properties.

    UR - http://www.scopus.com/inward/record.url?scp=78049528438&partnerID=8YFLogxK

    U2 - 10.1007/s00419-010-0435-6

    DO - 10.1007/s00419-010-0435-6

    M3 - Article

    VL - 80

    SP - 1197

    EP - 1212

    JO - Archive of Applied Mechanics

    JF - Archive of Applied Mechanics

    SN - 0939-1533

    IS - 10

    ER -