Field equations, equations of motion, and energy functionals for thick shells of revolution with arbitrary curvature and variable thickness from a three-dimensional theory

SummaryThis work uses tensor calculus to derive a complete set of three-dimensional field equations well-suited for determining the behavior of thick shells of revolution having arbitrary curvature and variable thickness. The material is assumed to be homogeneous, isotropic and linearly elastic. The equations are expressed in terms of coordinates tangent and normal to the shell middle surface. The relationships are combined to yield equations of motion in terms of orthogonal displacement components taken in the meridional, normal and circumferential directions. Strain energy and kinetic energy functionals are also presented. The equations of motion and energy functionals may be used to determine the static or dynamic displacements and stresses in shells of revolution, including free and forced vibration and wave propagation.

[1]  A. Leissa,et al.  Three-Dimensional Vibration Analysis of Thick, Complete Conical Shells , 2004 .

[2]  A. H. Shah,et al.  Three-Dimensional and Shell-Theory Analysis of Elastic Waves in a Hollow Sphere: Part 1—Analytical Foundation , 1969 .

[3]  S. Timoshenko,et al.  THEORY OF PLATES AND SHELLS , 1959 .

[4]  Jae-Hoon Kang,et al.  Three-Dimensional Vibration Analysis of Paraboloidal Shells , 2002 .

[5]  D. Gazis.  Three‐Dimensional Investigation of the Propagation of Waves in Hollow Circular Cylinders. I. Analytical Foundation , 1959 .

[6]  J. R. Hutchinson.  Transverse Vibrations of Beams, Exact Versus Approximate Solutions , 1981 .

[7]  Jae-Hoon Kang,et al.  Free vibrations of solid and hollow hemi-ellipsoids of revolution from a three-dimensional theory , 2004 .

[8]  E. Tsui.  Stresses in shells of revolution , 1968 .

[9]  Mitsuru Endo,et al.  Flexural Vibrations of a Ring with Arbitrary Cross Section , 1971 .

[10]  D. Gazis.  Three‐Dimensional Investigation of the Propagation of Waves in Hollow Circular Cylinders. II. Numerical Results , 1959 .

[11]  S. Raynor,et al.  Natural frequencies of finite circular cylinders in axially symmetric longitudinal vibration , 1971 .

[12]  O. G. McGee,et al.  A THREE‐DIMENSIONAL ANALYSIS OF THE SPHEROIDAL AND TOROIDAL ELASTIC VIBRATIONS OF THICK‐WALLED SPHERICAL BODIES OF REVOLUTION , 1997 .

[13]  K. Liew,et al.  Three-dimensional elasticity solutions for free vibrations of circular plates: A polynomials-Ritz analysis , 1999 .

[14]  K. Washizu.  Variational Methods in Elasticity and Plasticity , 1982 .

[15]  C. Chree.  The Equations of an Isotropic Elastic Solid in Polar and Cylindrical Co-ordinates their Solution and Application , 1889 .

[16]  A. Leissa,et al.  Response to ‘‘Comments on ‘Comparisons of vibration frequencies for rods and beams from one‐dimensional and three‐dimensional analyses’ ’’ [J. Acoust. Soc. Am. 100, 1890–1892 (1996)] , 1996 .

[17]  Arthur W. Leissa,et al.  THREE-DIMENSIONAL VIBRATIONS OF THICK CIRCULAR AND ANNULAR PLATES , 1998 .

[18]  AXISYMMETRIC VIBRATION OF LAMINATED ANNULAR PLATES COMPOSED OF TRANSVERSELY ISOTROPIC LAYERS , 1997 .

[19]  A. Leissa,et al.  Three-dimensional vibrations of hollow cones and cylinders with linear thickness variations , 1999 .

[20]  M. Poisson.  Mémoire sur l'équilibre et le mouvement des corps élastiques , 1828 .

[21]  Flexural Vibration of Unrestrained Cylinders and Disks , 1945 .

[22]  J. R. Hutchinson,et al.  Vibrations of Free Hollow Circular Cylinders , 1986 .

[23]  Horace Lamb,et al.  On the Vibrations of an Elastic Sphere , 1881 .

[24]  A. Gupta,et al.  Effect of secondary terms on axisymmetric vibration of circular plates , 1980 .

[25]  George Herrmann,et al.  Free Vibrations of Circular Cylindrical Shells , 1969 .

[26]  R. S. Girgis,et al.  Method for accurate determination of resonant frequencies and vibration behaviour of stators of electrical machines , 1981 .

[27]  Jae-Hoon Kang,et al.  Three-Dimensional Vibration Analysis of Thick Shells of Revolution , 1999 .

[28]  Arthur W. Leissa,et al.  Free Vibrations of Thick Hollow Circular Cylinders From Three-Dimensional Analysis , 1997 .

[29]  Ying-Te Lee,et al.  Axisymmetric vibration analysis of rotating annular plates by a 3D finite element , 2000 .

[30]  D. C. Gazis,et al.  Errata: Three‐Dimensional Investigation of the Propagation of Waves in Hollow Circular Cylinders. II [J. Acoust. Soc. Am. 31, 573–578 (1959)] , 1960 .

[31]  Jae-Hoon Kang,et al.  Three-dimensional vibrations of thick spherical shell segments with variable thickness , 2000 .

[32]  J. Radok,et al.  The theory of thin shells , 1959 .

[33]  G. M. L. Gladwell,et al.  Finite element analysis of the axisymmetric vibrations of cylinders , 1972 .

[34]  A. Leissa,et al.  Three-dimensional vibrations of thick, linearly tapered, annular plates , 1998 .

[35]  D. Haojiang,et al.  Nonaxisymmetric free vibrations of a spherically isotropic spherical shell embedded in an elastic medium , 1996 .

[36]  A. Leissa,et al.  Three-dimensional vibrations of solid cones with and without an axial circular cylindrical hole , 2004 .

[37]  G. Buchanan,et al.  Effect of Boundary Conditions on Free Vibration of Thick Isotropic Spherical Shells , 2002 .

[38]  N. Ganesan,et al.  Free vibration analysis of thick spherical shells , 1992 .

[39]  Philippe Young,et al.  NATURAL FREQUENCIES OF VIBRATION OF LAYERED HOLLOW SPHERES USING EXACT THREE-DIMENSIONAL ELASTICITY EQUATIONS , 1996 .

[40]  Determination op frequencies of natural vibrations of circular plates: PMM vol. 40, n≗1, 1976, pp. 112–119 , 1976 .

[41]  Y. Fung.  Foundations of solid mechanics , 1965 .

[42]  John D. Renton,et al.  Applied Elasticity: Matrix and Tensor Analysis of Elastic Continua , 2003 .

[43]  K. M. Liew,et al.  Vibration of Stress-Free Hollow Cylinders of Arbitrary Cross Section , 1995 .

[44]  G. Buchanan,et al.  Frequencies and mode shapes for thick truncated hollow cones , 2001 .

[45]  J. R. Hutchinson.  Axisymmetric Vibrations of a Free Finite‐Length Rod , 1972 .

[46]  K. Williams,et al.  A Theoretical and Experimental Study of Vibrations of Thick Circular Cylindrical Shells and Rings , 1988 .

[47]  Jae-Hoon Kang,et al.  Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation , 2003 .

[48]  W. Flügge,et al.  Tensor Analysis and Continuum Mechanics , 1972 .

[49]  J. R. Hutchinson,et al.  On the Vibration of Thick Annular Plates , 1987 .

[50]  Z. Celep.  Free vibration of some circular plates of arbitrary thickness , 1980 .

[51]  G. W. McHahon,et al.  A Finite Difference Analysis of the Vibrations of Solid Cylinders , 1967 .

[52]  Horace Lamb,et al.  On the Vibrations of a Spherical Shell , 1882 .

[53]  Z. Celep.  On the axially symmetric vibration of thick circular plates , 1978 .

[54]  A. Leissa,et al.  Response to ‘‘Comments on ‘Accurate vibration frequencies of circular cylinders from three‐dimensional analysis’ ’’ [J. Acoust. Soc. Am. 100, 1894–1895 (1996)] , 1996 .

[55]  A. Leissa,et al.  Vibration of shells , 1973 .

[56]  Y. Grigorenko,et al.  Analysis of the frequencies and modes of natural vibration of laminated hollow spheres in three- and two-dimensional formulations , 1989 .

[57]  K. Iyengar,et al.  Free vibration of circular plates of arbitrary thickness , 1978 .

[58]  Arvind H. Shah,et al.  Three-Dimensional and Shell-Theory Analysis of Elastic Waves in a Hollow Sphere: Part 2—Numerical Results , 1969 .

[59]  H. D. McNiven,et al.  Axially Symmetric Waves in Finite, Elastic Rods , 1962 .

[60]  G. Buchanan.  Vibration of Truncated Conical Cylinders of Crystal Class 6/mmm , 2000 .

[61]  Isaac Elishakoff,et al.  Refined Dynamical Theories of Beams, Plates and Shells and Their Applications: Proceedings of the Euromech-Colloquium 219 , 1987 .

[62]  Ya. M. Grigorenko,et al.  Analysis of the frequency characteristics of laminar cylindrical shells on the basis of different theories , 1984 .

[63]  K. Liew,et al.  Elasticity solutions for free vibrations of annular plates from three-dimensional analysis , 2000 .

[64]  A. Leissa,et al.  Comparisons of vibration frequencies for rods and beams from one‐dimensional and three‐dimensional analyses , 1995 .

[65]  H. Ding,et al.  Free Axisymmetric Vibration of Laminated Transversely Isotropic Annular Plates , 2000 .

[66]  Flexural Vibrations of a Thick-Walled Circular Cylinder According to the Exact Theory of Elasticity , 1960 .

[67]  Vibrations of a rigidly clamped circular plate , 1978 .

[68]  G. W. McMahon.  Experimental Study of the Vibrations of Solid, Isotropic, Elastic Cylinders , 1962 .

[69]  S. K. Datta,et al.  Three‐Dimensional Motion of Buried Pipeline. I: Analysis , 1986 .

[70]  Jae-Hoon Kang,et al.  Free Vibrations of Thick, Complete Conical Shells of Revolution From a Three-Dimensional Theory , 2005 .

[71]  Y. Lee,et al.  FINITE ELEMENT ANALYSIS OF THREE-DIMENSIONAL VIBRATIONS OF THICK CIRCULAR AND ANNULAR PLATES , 2000 .

[72]  Jae-Hoon Kang,et al.  Free vibration analysis of complete paraboloidal shells of revolution with variable thickness and solid paraboloids from a three-dimensional theory , 2005 .

[73]  I. S. Sokolnikoff.  Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua , 1964 .

[74]  Jae-Hoon Kang,et al.  Three-dimensional vibration analysis of thick, tapered rods and beams with circular cross-section , 2004 .

[75]  Arthur W. Leissa,et al.  Three-Dimensional Vibrations of Truncated Hollow Cones , 1995 .

[76]  J. R. Hutchinson.  Axisymmetric Flexural Vibrations of a Thick Free Circular Plate , 1979 .

[77]  J. R. Hutchinson.  Vibrations of Solid Cylinders , 1980 .

[78]  G. Gladwell,et al.  Natural frequencies of free finite-length circular cylinders , 1975 .

[79]  J. R. Hutchinson.  Vibrations of Thick Free Circular Plates, Exact Versus Approximate Solutions , 1984 .

[80]  H. Wang,et al.  VIBRATIONAL MODES OF THICK CYLINDERS OF FINITE LENGTH , 1996 .

[81]  A. Leissa,et al.  Three-Dimensional Vibrations of Thick, Circular Rings with Isosceles Trapezoidal and Triangular Cross-Sections , 2000 .

[82]  W. Ritz.  Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. , 1909 .

[83]  Jae-Hoon Kang,et al.  Three-dimensional vibration analysis of thick hyperboloidal shells of revolution , 2005 .

[84]  Arthur W. Leissa,et al.  Accurate vibration frequencies of circular cylinders from three‐dimensional analysis , 1995 .