Free Vibration of Orthotropic Skew Plates

1999 ◽  
Vol 122 (3) ◽  
pp. 313-317 ◽  
Author(s):  
A. M. Farag ◽  
A. S. Ashour
Keyword(s):  
Boundary Conditions ◽  
Transition Matrix ◽  
Longitudinal Direction ◽  
Displacement Function ◽  
Characteristic Matrix ◽  
Skew Angle ◽  
Kantorovich Method ◽  
Vibration Effect ◽  
Finite Strip Method ◽  
Semianalytical Method

The main purpose of this paper is to develop a fast converging semianalytical method for assessing the vibration effect on thin orthotropic skew (or parallelogram/oblique) plates. Since the geometry of the skew plate is not helpful in the mathematical treatments, the analysis is often performed by more complicated and laborious methods. A successive conjunction of the Kantorovich method and the transition matrix is exploited herein to develop a new modification of the finite strip method to reduce the complexity of the problem. The displacement function is expressed as the product of a basic trigonometric series function in the longitudinal direction and an unknown function that has to be determined in the other direction. Using the new transition matrix, after necessary simplification and the satisfaction of the boundary conditions, yields a set of simultaneous equations that leads to the characteristic matrix of vibration. The influence of the skew angle, the aspect ratio, the properties of orthotropy, and the prescribed boundary conditions are investigated. Convergence of the solution is investigated and the accuracy of the results is compared with that available from other numerical methods. The numerical results show that the convergence is rapidly deduced and the comparisons agree very well with known results. [S0739-3717(00)00202-6]

Buckling of Circular Cylindrical Shells Using a Displacement Function

1974 ◽  
Vol 96 (4) ◽  
pp. 1322-1327
Author(s):  
Shun Cheng ◽  
C. K. Chang
Keyword(s):  
Boundary Conditions ◽  
Axial Compression ◽  
Cylindrical Shells ◽  
External Pressure ◽  
Displacement Function ◽  
Combined Loading ◽  
Trial Solution ◽  
Governing Differential Equation ◽  
Circular Cylindrical Shells ◽  
The Stability

The buckling problem of circular cylindrical shells under axial compression, external pressure, and torsion is investigated using a displacement function φ. A governing differential equation for the stability of thin cylindrical shells under combined loading of axial compression, external pressure, and torsion is derived. A method for the solutions of this equation is also presented. The advantage in using the present equation over the customary three differential equations for displacements is that only one trial solution is needed in solving the buckling problems as shown in the paper. Four possible combinations of boundary conditions for a simply supported edge are treated. The case of a cylinder under axial compression is carried out in detail. For two types of simple supported boundary conditions, SS1 and SS2, the minimum critical axial buckling stress is found to be 43.5 percent of the well-known classical value Eh/R3(1−ν2) against the 50 percent of the classical value presently known.

An Improved Fourier–Ritz Method for Analyzing In-Plane Free Vibration of Sectorial Plates

2017 ◽  
Vol 84 (9) ◽  
Author(s):  
Siyuan Bao ◽  
Shuodao Wang ◽  
Bo Wang
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Displacement Function ◽  
Accurate Solution ◽  
Previous Method ◽  
Fourier Series Expansion ◽  
Arbitrary Boundary ◽  
Fourier Cosine ◽  
Radial Variable

A modified Fourier–Ritz approach is developed in this study to analyze the free in-plane vibration of orthotropic annular sector plates with general boundary conditions. In this approach, two auxiliary sine functions are added to the standard Fourier cosine series to obtain a robust function set. The introduction of a logarithmic radial variable simplifies the expressions of total energy and the Lagrangian function. The improved Fourier expansion based on the new variable eliminates all the potential discontinuities of the original displacement function and its derivatives in the entire domain and effectively improves the convergence of the results. The radial and circumferential displacements are formulated with the modified Fourier series expansion, and the arbitrary boundary conditions are simulated by the artificial boundary spring technique. The number of terms in the truncated Fourier series and the appropriate value of the boundary spring retraining stiffness are discussed. The developed Ritz procedure is used to obtain accurate solution with adequately smooth displacement field in the entire solution domain. Numerical examples involving plates with various boundary conditions demonstrate the robustness, precision, and versatility of this method. The method developed here is found to be computationally economic compared with the previous method that does not adopt the logarithmic radial variable.

A new Fourier series method and displacement function for in-plane vibration and power flow characteristics analysis of isotropic rectangular plates

2019 ◽  
Vol 50 (6) ◽  
pp. 176-194
Author(s):  
Kavikant Mahapatra ◽  
SK Panigrahi
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Power Flow ◽  
Flow Characteristics ◽  
Displacement Function ◽  
Rectangular Plates ◽  
Series Method ◽  
Plane Vibration ◽  
Fourier Series Method ◽  
Beam Displacement

The generation of in-plane vibration in plates is an important issue and frequently occurs due to the presence of excitations in the ship’s hull due to turbulent fluid flows, turbulent airflow excitation on aerospace structures, gear system subjected to axial excitation, assemblies housing piezoelectric crystals and sandwiched plates, and so on. The present analysis aims to establish a universal and numerically efficient method for determination of in-plane vibration characteristics of isotropic rectangular plates both for conventional and general boundary conditions. The new in-plane Fourier series and displacement function of the plate have been developed using beam displacement functions in x and y directions, respectively, under in-plane condition. A modified Fourier series assumption for the in-plane beam displacement has been utilised and further developed as plate displacement function. The computational efficiency of the present method is compared in terms of convergence of natural frequency parameter, speed of execution and manual convenience to reduce human errors with the frequently used Fourier series method by various researchers. Rayleigh–Ritz procedure has been applied to determine the in-plane natural frequencies. The mode shapes for few conventional and generally varying boundary conditions have been presented and analysed. The dynamic response has been obtained and analysed in terms of the in-plane mobility and power flow characteristics of the plate under varying boundary conditions. The validity of results obtained by the current method has shown excellent accuracy and faster convergence with the existing results. The present results can provide a benchmark to analyse the dynamic in-plane response of plate systems being used for built-up structures in real engineering applications.

Stability of laminated composite and sandwich FGM shells using a novel isogeometric finite strip method

2019 ◽  
Vol 37 (4) ◽  
pp. 1369-1395 ◽  
Author(s):  
Mohammad Amin Shahmohammadi ◽  
Mojtaba Azhari ◽  
Mohammad Mehdi Saadatpour ◽  
Saeid Sarrami-Foroushani
Keyword(s):  
Boundary Conditions ◽  
Critical Loads ◽  
Laminated Composite ◽  
Functionally Graded ◽  
Laminated Shells ◽  
Content Type ◽  
Finite Strip ◽  
Strip Method ◽  
Finite Strip Method ◽  
The Stability

Purpose This paper aims to analyze the stability of laminated shells subjected to axial loads or external pressure with considering various geometries and boundary conditions. The main aim of the present study is developing an efficient combined method which uses the advantages of different methods, such as finite element method (FEM) and isogeometric analysis (IGA), to achieve multipurpose targets. Two types of material including laminated composite and sandwich functionally graded material are considered. Design/methodology/approach A novel type of finite strip method called isogeometric B3-spline finite strip method (IG-SFSM) is used to solve the eigenvalue buckling problem. IG-SFSM uses B3-spline basis functions to interpolate the buckling displacements and mapping operations in the longitudinal direction of the strips, whereas the Lagrangian functions are used in transverse direction. The current presented IG-SFSM is formulated based on the degenerated shell method. Findings The buckling behavior of laminated shells is discussed by solving several examples corresponding to shells with various geometries, boundary conditions and material properties. The effects of mechanical and geometrical properties on critical loads of shells are investigated using the related results obtained by IG-SFSM. Originality/value This paper shows that the proposed IG-SFSM leads to the critical loads with an approved accuracy comparing with the same examples extracted from the literature. Moreover, it leads to a high level of convergence rate and low cost of solving the stability problems in comparison to the FEM.

FREE VIBRATION ANALYSIS OF DISCRETELY STIFFENED SKEW PLATES

2001 ◽  
Vol 01 (01) ◽  
pp. 125-144 ◽  
Author(s):  
HUAN ZENG ◽  
CHARLES W. BERT
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Vibration Characteristics ◽  
Skew Angle ◽  
Various Boundary Conditions ◽  
Skew Plate ◽  
Convergence Study

Stiffened skew plates find application in various engineering fields. The free vibration characteristics of such plates have been studied by various methods. An orthogonally stiffened skew plate is a skew plate with stiffeners running orthogonal to two opposite edges. To the best knowledge of the present investigators, no previous work has been done for free vibration characteristics of skew plates of such stiffening geometry. The present work studies the free vibration of such plates. The pb-2 Rayleigh–Ritz method was employed due to its accuracy and computational efficiency. The conventional finite element method was also used as a comparative check. A convergence study was first performed for various boundary conditions. Then the vibration of orthogonally stiffened skew plates with different boundary conditions was studied. Close agreement was found between these two methods. The variations of natural frequencies with different parameters, including skew angle ϕ, edge ratio b/a, and height-thickness ratio f/h, were investigated for three types of boundary conditions.

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