Free Vibration of Curvilinearly Stiffened Shallow Shells

2015 ◽  
Vol 137 (3) ◽  
Author(s):  
Peng Shi ◽  
Rakesh K. Kapania ◽  
C. Y. Dong
Keyword(s):  
Free Vibration ◽  
Shell Theory ◽  
Ritz Method ◽  
Three Dimensional ◽  
Beam Theory ◽  
Shell Element ◽  
Shallow Shells ◽  
First Order ◽  
Stiffened Shells ◽  
Good Agreement

The free vibration of curvilinearly stiffened shallow shells is investigated by the Ritz method. Based on the first-order shear deformation shell theory and three-dimensional (3D) curved beam theory, the strain and kinetic energies of the stiffened shells are introduced. The stiffener can be placed anywhere within the shell, without the need for having the stiffener and shell element nodes coincide. Numerical results with different geometrical shells and boundary conditions and different stiffener locations and curvatures are analyzed to verify the feasibility of the presented Ritz method for solving the problems. The results show good agreement with those using other methods, e.g., using a converged set of results obtained by Nastran.

Free Vibration of Curvilinearly Stiffened Shallow Shells

2013 ◽  
Author(s):  
Peng Shi ◽  
Rakesh K. Kapania
Keyword(s):  
Free Vibration ◽  
Ritz Method ◽  
Beam Theory ◽  
Fe Method ◽  
Shallow Shells ◽  
First Order ◽  
Stiffened Shells ◽  
Method Base ◽  
Doubly Curved ◽  
Good Agreement

The free vibration of curvilinearly stiffened doubly curved shallow shells is investigated by the Ritz method. Base on the first order shear deformation shell theory and Timoshenko’s 3-D curved beam theory, the strain and kinetic energies of the stiffened shells are introduced. Numerical results with different geometrical shells and boundary conditions, and different stiffener locations and curvatures are analyzed to verify the feasibility of the presented Ritz method for solving the problems. The results show good agreement with those using the FE method.

3D Vibration Analysis of Hermetic Capsules by Using Ritz Method

2017 ◽  
Vol 17 (03) ◽  
pp. 1750040
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Eigenvalue Problem ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Legendre Polynomials ◽  
Present Method ◽  
Ritz Method ◽  
Three Dimensional ◽  
Dynamic Equations ◽  
Vibration Frequencies ◽  
Good Agreement

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of a hermetic capsule comprising a cylinder closed with hemi-ellipsoidal caps at both ends. Unlike conventional shell theories, which are mathematically 2D, the present method is based upon the 3D dynamic equations of elasticity. Displacement components [Formula: see text], [Formula: see text], and [Formula: see text] in the radial, circumferential, and axial directions, respectively, are taken to be periodic in [Formula: see text] and in time, and the Legendre polynomials in the r and z directions instead of ordinary ones. Potential (strain) and kinetic energies of the hermetic capsule are formulated, and the Ritz method is used to solve the eigenvalue problem, thereby yielding upper bound values of the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Typical convergence studies are carried out for the first five frequencies. The frequencies from the present 3D method are in good agreement with those obtained from other 3D approach and 2D shell theories proposed by previous researchers.

scholarly journals A Semianalytical Approach for Free Vibration Characteristics of Functionally Graded Spherical Shell Based on First-Order Shear Deformation Theory

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Fuzhen Pang ◽  
Cong Gao ◽  
Jie Cui ◽  
Yi Ren ◽  
Haichao Li ◽  
...  
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Numerical Verification ◽  
First Order

This paper describes a unified solution to investigate free vibration solutions of functionally graded (FG) spherical shell with general boundary restraints. The analytical model is established based on the first-order shear deformation theory, and the material varies uniformly along the thickness of FG spherical shell which is divided into several sections along the meridian direction. The displacement functions along circumferential and axial direction are, respectively, composed by Fourier series and Jacobi polynomial regardless of boundary restraints. The boundary restraints of FG spherical shell can be easily simulated according to penalty method of spring stiffness technique, and the vibration solutions are obtained by Rayleigh–Ritz method. To verify the reliability and accuracy of the present solutions, the convergence and numerical verification have been conducted about different boundary parameters, Jacobi parameter, etc. The results obtained by the present method closely agree with those obtained from the published literatures, experiments, and finite element method (FEM). The impacts of geometric dimensions and boundary conditions on the vibration characteristics of FG spherical shell structure are also presented.

Vibration of Shallow Shells: A Review With Bibliography

1997 ◽  
Vol 50 (8) ◽  
pp. 431-444 ◽  
Author(s):  
K. M. Liew ◽  
C. W. Lim ◽  
S. Kitipornchai
Keyword(s):  
Thin Shell ◽  
Shell Theory ◽  
Free Vibration Analysis ◽  
Review Article ◽  
Shallow Shells ◽  
First Order ◽  
Recent Developments ◽  
Thick Shells ◽  
Thin Shell Theory ◽  
Introductory Review

This review article documents recent developments in the free vibration analysis of thin, moderately thick, and thick shallow shells. An introductory review of the studies in Kirchhoff-Love classical thin shell theory is given. The development of studies in moderately thick shells incorporating the effects of transverse shear deformation and rotary inertia is detailed. This review article mainly focuses on research advances in vibration studies since the 1970s using the classical Kirchhoff-Love, first-order, and higher-order theories. The validity and range of applicability of these theories are examined. There are 163 references listed at the end of the article.

Three-Dimensional Vibration Analysis of Thick, Complete Conical Shells

2004 ◽  
Vol 71 (4) ◽  
pp. 502-507 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Present Method ◽  
Shell Theory ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Algebraic Polynomials ◽  
Shells Of Revolution ◽  
Conical Shells ◽  
Thin Shell Theory ◽  
Time Periodic

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components ur,uz, and uθ in the radial, axial, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z-directions. Potential (strain) and kinetic energies of the conical shells are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the conical shells. Novel numerical results are presented for thick, complete conical shells of revolution based upon the 3D theory. Comparisons are also made between the frequencies from the present 3D Ritz method and a 2D thin shell theory.

scholarly journals Numerical Investigation on Steel Square HSS Columns Strengthened with Polymer-mortar

2019 ◽  
pp. 1-25
Author(s):  
Khaled M. El-Sayed ◽  
Ahmed S. Debaiky ◽  
Nader N. Khalil ◽  
Ibrahim M. El-Shenawy
Keyword(s):  
Three Dimensional ◽  
Shell Element ◽  
Fe Analysis ◽  
Experimental Results ◽  
Fe Model ◽  
Cross Sectional ◽  
Axial Strength ◽  
Polymer Mortar ◽  
Slender Columns ◽  
Good Agreement

This paper presents the results of finite element (FE) analysis of axially loaded square hollow structural steel (HSS) columns, strengthened with polymer-mortar materials. Three-dimensional nonlinear FE model of HSS slender columns were developed using thin-shell element, considering geometric and material nonlinearity. The polymer-mortar strengthening layer was incorporated using additional layers of the shell element. The FE model has been performed and then verified against experimental results obtained by the authors [1]. Good agreement was observed between FE analysis and experimental results. The model was then used in an extended parametric study to examine selected AISC square HSS columns with different cross-sectional geometries, slenderness ratios, thicknesses of mortar strengthening layer, overall geometric imperfections, and level of residual stresses. The effectiveness of polymer-mortar in increasing the column’s axial strength is observed. The study also demonstrated that polymer-mortar strengthening materials is more effective for higher slenderness ratios. An equivalent steel thickness is also accounted for the mortar strengthened HSS columns to discuss the effectiveness of polymer-mortar strengthening system. The polymer-mortar strengthening system is more effective for HSS columns with higher levels of out-of-straightness. Level of residual stress has a slight effect on the gain in the column’s axial strength strengthened with polymer-mortar.

scholarly journals Application of alternative II theory to vibration and stability analysis of thick rectangular plates (Isotropic and orthotropic)

2020 ◽  
Vol 39 (1) ◽  
pp. 52-62
Author(s):  
O.M. Ibearugbulem ◽  
S.I. Ebirim ◽  
U.C. Anya ◽  
L.O. Ettu
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Natural Frequency ◽  
Deformation Theory ◽  
Thick Plate ◽  
Shape Function ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
First Order ◽  
Fundamental Natural Frequency

This work analysed the free vibration and stability of thick isotropic and orthotropic plates with SSSS and SSFS support conditions by applying the alternative II theory based on polynomial shape function. The total potential energy which was obtained by combining the strain energy and external work was reduced to three governing equations using Ritz method. Polynomial shape function which varies with Poisson’s ratio was substituted into the governing equation to obtain the fundamental natural frequency, linear frequency and critical buckling load. The values of frequencies of the first mode and critical loads obtained were compared with those obtained using first order shear deformation theory. For span depth ratio of 10, the fundamental linear frequency for orthotropic SSFS plate corresponding to modulus of elasticity ratios (E1/E2) of 10, 25 and 40 are 0.00156, 0.00219 and 0.00255Hz. The corresponding values using first order shear deformation theory are 0.00152, 0.00212 and 0.00245Hz. Keywords: Fundamental natural frequency, SSSS plate, SSFS plate, Ritz method, Orthotropic thick plate, Isotropic thick plate, Stability, Free vibration

scholarly journals Accurate Results for Free Vibration of Doubly Curved　Shallow Shells of Rectangular Planform (Part.1)

2021 ◽  
Vol 4 (1) ◽  
pp. 29-36
Author(s):  
Daisuke Narita ◽  
Yoshihiro Narita
Keyword(s):  
Free Vibration ◽  
Shell Theory ◽  
Frequency Equation ◽  
Shell Shape ◽  
Comparison Study ◽  
Shallow Shells ◽  
Wide Range ◽  
Out Of Plane ◽  
Edge Conditions ◽  
Doubly Curved

A method is presented for determining the free vibration frequencies of doubly curved, isotropic shallow shells under general edge conditions and is used to obtain accurate natural frequencies for wide range of geometric parameters. Based on the shallow shell theory applicable to thin thickness shells, a method of Ritz is extended to derive a frequency equation wherein the displacement functions are modified to accommodate arbitrary sets of edge conditions for both in-plane and out-of-plane motions. In numerical computation, convergence is tested against series terms and comparison study is made with existing results by other authors. Twenty one sets of frequency parameters are tabulated for a wide range of shell shape and curvature ratio to serve as data for future comparison and practical design purpose.  

scholarly journals Isogeometric iFEM Analysis of Thin Shell Structures

Sensors ◽  
2020 ◽  
Vol 20 (9) ◽  
pp. 2685 ◽  
Author(s):  
Adnan Kefal ◽  
Erkan Oterkus
Keyword(s):  
Large Scale ◽  
Shell Theory ◽  
Weighted Least Squares ◽  
Three Dimensional ◽  
Shell Element ◽  
Surface Strain ◽  
Curvilinear Coordinates ◽  
Shell Structures ◽  
Engineering Structures ◽  
Shape Sensing

Shape sensing is one of most crucial components of typical structural health monitoring systems and has become a promising technology for future large-scale engineering structures to achieve significant improvement in their safety, reliability, and affordability. The inverse finite element method (iFEM) is an innovative shape-sensing technique that was introduced to perform three-dimensional displacement reconstruction of structures using in situ surface strain measurements. Moreover, isogeometric analysis (IGA) presents smooth function spaces such as non-uniform rational basis splines (NURBS), to numerically solve a number of engineering problems, and recently received a great deal of attention from both academy and industry. In this study, we propose a novel “isogeometric iFEM approach” for the shape sensing of thin and curved shell structures, through coupling the NURBS-based IGA together with the iFEM methodology. The main aim is to represent exact computational geometry, simplify mesh refinement, use smooth basis/shape functions, and allocate a lower number of strain sensors for shape sensing. For numerical implementation, a rotation-free isogeometric inverse-shell element (isogeometric Kirchhoff–Love inverse-shell element (iKLS)) is developed by utilizing the kinematics of the Kirchhoff–Love shell theory in convected curvilinear coordinates. Therefore, the isogeometric iFEM methodology presented herein minimizes a weighted-least-squares functional that uses membrane and bending section strains, consistent with the classical shell theory. Various validation and demonstration cases are presented, including Scordelis–Lo roof, pinched hemisphere, and partly clamped hyperbolic paraboloid. Finally, the effect of sensor locations, number of sensors, and the discretization of the geometry on solution accuracy is examined and the high accuracy and practical aspects of isogeometric iFEM analysis for linear/nonlinear shape sensing of curved shells are clearly demonstrated.

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