Ritz Method

The first goal of this work is to present a literature review regarding the use of several sets of admissible functions in the Ritz method. The papers reviewed deal mainly with the analysis of buckling and free vibration of isotropic and anisotropic beams and plates. Theoretically,
in order to obtain a correct solution, the set of admissible functions must not violate the essential or geometric boundary conditions and should also be linearly independent and complete. However, in practice, some of the sets of functions proposed in the literature present a bad numerical
behavior, namely in terms of convergence, computational time and stability. Thus, a second goal of the present work is to compare the performance of several sets of functions in terms of these three features. To achieve this objective, the free vibration analysis of a fully clamped rectangular plate is carried out using six different sets of functions, along with the study of the convergence of natural frequencies and mode shapes, the computational time and the
numerical stability.

>> syms A0 Uz Ka Ge E I p p2 x y z TP pi eqn tur tur2 tur3 >> >> %VERİLENLER >> Uz = 6; % uzunluk [m] >> y = 3;% kirişin orta noktası (Uz/2) [m] >> Ge = 0.25; % genislik [m] >> Ka = 0.5; % kalınlık [m] >> E = 30*10^6 ; % elastite modülü [kN/m^2] >> I = Ge*Ka^3/12 ;%dikdörtgen kesitli kirisin atalet momenti; %genislik * kalınlık^3 /12 [m^4] >> p = 40; % kiriş üzerindeki yük [kN/m] >> p2=-40; %koordinat sisteminde ters yönde olduğundan p=-p2 [kN/m] >> u(x)= A0*sin(x*pi / Uz) % yer değiştirme fonksiyonu u(x) = A0*sin((pi*x)/6) >> >> >> %RİTZ METODU İLE ÇÖZÜM >> >> %Mesnetlerde çökme olmaması gerekir.Kontrol ediyoruz. >> %Mesnetlerin başlangıç ve bitim noktası değerlerini yerine koyuyoruz. >> >> u(0)%mesnetlerde çökme olmaması gerekir oyüzden x=0 kontrol ediyoruz 11

A numerical model for the analysis of the free vibration of muttilayered, fully free rectangular plates with arbitrary layups is proposed. The model is based on a displacement field that assumes a nonlinear distribution of the in-plane displacements with respect to the thickness coordinate and a transverse displacement that is constant through the plate thickness. The displacement field accountsfora parabolic distribution of'the transverse shear strains and satisfies the condition of zero transverse shear stresses on the top and bottom surfaces of the plate. The mathematical problem is solved by the Ritz technique, which relies on the minimization of the maximum Lagrangian functional, using a displacement field defined by beam functions in the form of finite series with undetermined parameters. The required mathematical operations for the development of the eigenvalue problem are carried out by symbolic manipulation. By deleting the higher-order terms of Reddy's displacement field or assuming the Love-Kirchhoff hypothesis, the first-order shear deformation and the classical plate theory models, respectively, are also obtained. These models are used to analyze several laminated composite plate problems, and the eigenfrequency results are compared among them and with alternative solutions.

This paper presents a method to predict the through-thickness moisture content distribution and associated induced deformations of paper and cardboard sheets as they are subjected to relative humidity changes. The transient moisture diffusion problem is solved using a “natural” analytic approach that has previously been applied for solving transient heat conduction in multi-layer solids. The deformation behaviour of the sheet during the moisture diffusion process is predicted using a semi-analytical approach based on a Rayleigh–Ritz minimization of the total potential energy. Geometrically nonlinear effects are taken into account. Curvatures of the originally flat sheet are predicted as a function of time, as are the shapes of the sheet for steady-state condition. As multiple solutions exist, stability is studied. The developed model was used to study the deformation behaviour of one paper and two cardboard sheets. Comparisons with finite-element results demonstrate that the develop...

The natural frequencies of skewed cantilevered thick plates are determined by using the Ritz method. The present work is the first known three-dimensional study of the problem. Assumed displacement functions are in the form of algebraic polynomials which satisfy the fixed face conditions exactly, and which are mathematically complete. Accurate natural frequencies are calculated for skewed thick plates having arbitrary degrees of skewness. Detailed numerical studies reveal interesting trends concerning the variation of frequencies with increasing skew angle. Results obtained by using the present method are compared with those obtained by using three-dimensional finite elements.

This thesis investigates the linear free vibrations of functionally graded Carbon Nano-tube reinforced Composite (FG-CNTRC) beams using Slender (Euler-Bernoulli’s) beam theory. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction and estimated through the rule of mixture. The Ritz method is employed to derive the governing Eigen value equation which is then solved by a direct iterative method to obtain the linear frequencies of FG-CNTRC beams with H-H Supports. A detailed parametric study is conducted to study the influences of Nanotube volume fraction, vibration amplitude, and slenderness ratio on the linear free vibration characteristics of FG-CNTRC beams.

In this study, the dynamic buckling of an elastic cylindrical shell with variable thickness, subject to a uniform external pressure which is a power function of time, has been considered. Initially, the fundamental relations and Donnell-type dynamic buckling equation of an elastic cylindrical shell with variable thickness have been obtained. Then, employing Galerkin's method, these equations have been reduced to a time-dependent differential equation with variable coefficients. Finally, applying a special Ritz-type method, the critical static and dynamic loads, the corresponding wave numbers and dynamic factor have been found analytically. Using those results, the effects of the variation of the thickness with a linear, a parabolic or an exponential function in the axial direction and the effect of the variation of the power of time in the external pressure expression are studied using pertinent computations. It is observed that these effects change appreciably the critical parameters of the problem. The present method has been verified, comparing the results of the present work and those of previous works in the literature, for a shell with constant thickness subject to a uniform external pressure varying linearly with time.