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Design and applications
Published in Peter Domone, John Illston, Construction Materials, 2018
Large-deflection theory solutions to the bending of plates are more complex to calculate but some design codes, such as ASTM E-1300, provide reference charts to save the designer lengthy calculations. Some other codes use simple bending calculations but factor up the design stress to allow for the fact that the calculated stress will be overestimated. This is one of the reasons why design stresses from window codes cannot be taken to be safe values for more general application to glass engineering.
Finite Elements for Beams
Published in Irving H. Shames, Clive L. Dym, Energy and Finite Element Methods in Structural Mechanics, 2018
Irving H. Shames, Clive L. Dym
When such continuity of deformation between elements is possible, we say that we have conformable elements. In more complex structures such as plates and torsion rods we may not choose to use comformable elements to work with, for reasons to be discussed later. In such cases convergence criteria for the finite element process as the size of the element is decreased are very important. That is, achieving good results requires additional considerations beyond those needed for truss and beam problems. Hence, before we can proceed to more complex problems, we must pause to present in some detail such basic considerations as the choice of elements, consideration of the appropriate number of degrees of freedom for an element, finding of proper interpolation functions, and the important questions of the convergence requirements for the finite element process. We consider these items and more in the following chapter. We can then move to plane stress, the bending of plates, torsion, etc., as well as non-solid-mechanical applications of the finite element process.
Two-Dimensional Stress-Deformation Analysis
Published in Chandrakant S. Desai, Tribikram Kundu, Introductory Finite Element Method, 2017
Chandrakant S. Desai, Tribikram Kundu
Most real problems are three-dimensional. Under certain assumptions, which can depend on the geometrical and loading characteristics, it is possible to approximate many of them as two-dimensional. Such two-dimensional approximations generally involve two categories: plane deformations and bending deformations. In the case of plane deformation, we encounter subcategories such as plane stress, plane strain, and axisymmetric; in the case of bending, we deal with problems such as bending of plates, slabs, and pavements.
Investigating the free vibration of viscoelastic FGM Timoshenko nanobeams resting on viscoelastic foundations with the shear correction factor using finite element method
Published in Mechanics Based Design of Structures and Machines, 2022
Ghali Drici, Ismail Mechab, Hichem Abbad, Noureddine Elmeiche, Belaid Mechab
Furthermore, it is worth emphasizing that the mechanical behavior of elastic foundations has already been studied by (Winkler 1867). However, Pasternak (1954) presented a two-parameter model that takes into account the shear strain (rotational stiffness) and the vertical stiffness as well; he then compared it with the one-parameter model. It should be mentioned that Winkler’s model can be considered as a special case of Pasternak’s model when the shear modulus is set to zero. A literature review showed that few studies, based on the nonlocal shear deformation beam theory, have been carried out to study the free vibration of rectangular FGM nanobeams. It is useful to know that the classical Love-Kirchhoff plate theory to study the bending of plates, also known as the classical thin plate theory (CPT), does not take into account the effects of shear deformation.
Size-dependent free vibration analysis of Mindlin nano-plates with curvilinear plan-forms by a high order curved hierarchical finite element
Published in Mechanics of Advanced Materials and Structures, 2020
A. Necira, S. A. Belalia, A. Boukhalfa
A number of studies in recent years dealing with nano-scaled beams and plates via nonlocal continuum have been published. There are already different studies on the use of nonlocal continuum modeling in vibration and static analysis of carbon nanotubes (CNTs) and nano-structures, by Wang et al. [5], [6], respectively. Shen and Zhang [7] reported buckling of double-walled carbon nanotubes (CNTs), application of nonlocal theory for beam vibration [8]. Reddy [9] formulated the nonlocal nonlinear models for bending of classical and shear deformation theories of beams and plates, variational formulation, and finite element analysis of nanobeams and nanoplates [10]. Ghorbanpour Arani et al. [11] applied the nonlocal Mindlin plate theory to investigate the buckling smart control of a single layer graphene sheet (SLGS). In their study, SLGS is elastically coupled with polyvinylidene fluoride (PVDF) nanoplate and controlled by applying external electric voltage in thickness direction of PVDF nanoplate. Study of the buckling analysis of graphene sheet via nonlocal Mindlin plate theory [12], [13] and vibration analysis of multilayered graphene sheets (MLGS) [14], nonlocal plate model for vibration of SLGS [15], [16]. Ghorbanpour Arani and Haghparast [17] applied the sinusoidal shear deformation theory (SSDT) to investigate the size-dependent vibration of axially moving viscoelastic microplate by means of the modified couple stress theory (MCST) which contains an internal material length scale parameter related to the material microstructures. In their study, Kelvin's model was utilized to obtain viscoelastic properties of the micro-plate. Based on nonlocal Mindlin plate theory, the vibration characteristics of an embedded nanoplate-based nanoelectromechanical sensor made of PVDF carrying a nanoparticle with different masses at any position were investigated by Haghshenas and Ghorbanpour Arani [18]. Arefi and Zenkour [19] applied the Hamilton's principle, Kirchhoff plate theory, and nonlocalelectro-magneto-elastic relations to analyze the free vibration problem of a sandwich nanoplate rest on visco-Pasternak's foundation. Assadi [20] investigated the effect of surface properties on forced vibration of rectangular nanoplates through an analytical method. Aghababaei and Reddy [21] applied the nonlocal third-order shear deformation plate theory to study the vibration and bending of plates, CPT, and Mindlin nonlocal theory for plate vibration [22], [23], Aksencer and Aydogdu [24] investigated the effect of small scale and different boundary conditions on vibration and buckling of nonlocal plates by the Levy method. Jomehzadeh and Saidi [25] studied three-dimensional vibration of nanoplates using nonlocal continuum mechanics. Malekzadeh et al. [26] applied the differential quadrature method (DQM), Eringen's nonlocal elasticity theory, and first-order shear deformation plate theory (FSDT) to analyze the free vibration problem of quadrilateral nanoplates.