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Plate Bending by Approximate and Numerical Methods
Published in Eduard Ventsel, Theodor Krauthammer, Thin Plates and Shells, 2001
Eduard Ventsel, Theodor Krauthammer
We are now coming to the presentation of the direct methods. We begin with a study of the Ritz method. The Ritz method belongs among the so-called variational methods that are commonly used as approximate methods for a solution of various boundary value problems of mechanics. These methods are based on variational principles of mechanics discussed in Sec. 2.6.
The Classical Beam Theory
Published in J. N. Reddy, Theories and Analyses of Beams and Axisymmetric Circular Plates, 2022
The Ritz method applies to all problems, linear or nonlinear, as long as one can cast the governing equations in the form of a variational statement [see Eqs. (3.9.6a) and (3.9.6b)]. The procedure to develop variational forms, also called weak forms, from governing differential equations arising in any field, can be found in [9, 155].
Finite Element Analysis: Background Concepts
Published in Steven M. Lepi, Practical Guide to Finite Elements, 2020
Using the Ritz method, the variational calculus problem of minimizing a functional is replaced by the less complicated, approximate, differential calculus problem of minimizing a function. For example, consider again the steering link problem, illustrated once more in Figure 1.17.
On linear and nonlinear bending of functionally graded graphene nanoplatelet reinforced composite beams using Gram-Schmidt-Ritz method
Published in Mechanics Based Design of Structures and Machines, 2023
Wachirawit Songsuwan, Chamlong Prabkeao, Nuttawit Wattanasakulpong
However, the previous investigations of FG-GPLRC structures described above are assumed to be in the form of multiple-layer distributions of GPLs. To avoid the stress concentration between the layers, this current study supposes that the GPLs are continuously distributed in the polymer matrix by using the reinforcing functions (Wang et al. 2019a, 2019b; Wang, Xie, et al. 2020; Wang, Xie, et al. 2020). For this type of FG-GPLRC structure having a continuous distribution of GPLs, the volume and weight fractions of GPLs are controlled by the smooth functions in any designed direction. In the literature survey, there is a very limited amount of research dealing with the mechanical behavior of such structures. In terms of numerical methodologies used for solving mechanical problems of engineering structures, the Ritz method is one of the popular methods that can provide accurate solutions for structures with arbitrary boundary conditions. This method was successfully used to solve various problems in engineering fields (Akbaş et al. 2021). The Gram-Schmidt procedure can be used to generate the displacement functions for the Ritz method. This procedure allows us to obtain numerically stable functions. Therefore, the Gram-Schmidt-Ritz method can provide high spectral accuracy and fast convergence (Nallim, Martinez, and Grossi 2005; Nallim and Oller 2008; Rango, Bellomo, and Nallim 2015; Chaikittiratana and Wattanasakulpong 2020).
The Kantorovich method applied to bending, buckling, vibration, and 3D stress analyses of plates: A literature review
Published in Mechanics of Advanced Materials and Structures, 2019
Pairod Singhatanadgid, Thanyarat Singhanart
The Ritz and Galerkin methods are very similar. In some problems, the Ritz and Galerkin methods yield identical solutions. If the Galerkin equation is derived based on the variational principle, the algebraic equations produced by the Galerkin method are identical to those of the Ritz method [2]. The Ritz method applied to bending, buckling, and vibration of plates is based on the principle of minimum potential energy. On the other hand, the Galerkin equations may be derived from the equilibrium equation or the energy principle. The application of the Ritz and Galerkin methods to structural problems is outlined thoroughly in Refs. [2, 4, 36–38]. In summary, the Ritz method is a variational method that is usually employed as an approximate method for boundary value problems in mechanics. For a plate structure, the out-of-plane displacement of the specimen is usually approximated by a series in the form of where fi(x) and gi(y) are basis functions that satisfy the geometrical boundary conditions of the plate, and Ci are unknown coefficients to be determined. These coefficients are determined by substituting the assumed function in Eq. (1) into the total potential energy Π of the plate. After performing the integration over the domain of the problem, the total potential energy can be written as a function of the unknown coefficients Ci, i.e., Π = Π(C1, C2, …CN). The principle of minimum potential energy requires that