Calculation of clamped plates in the formation of the three-dimensional problem of the theory of elasticity

Lisitsyn, B. M.

doi:10.1007/bf00901836

Cited by 5 publications

(3 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here, elements of slab volume dVp and surface dS are introduced, as well as limit values of Naghdi approximation [18], used in determination of compression deformations: 𝑊 ̄(𝑥, у, 𝑧) ≃ 𝑤(𝑥, 𝑦) + 𝑧 2 𝑤 2 (𝑥, 𝑦), where 𝑞 ±corresponding values of the external load on the external surfaces 𝑧 = ±ℎ of the slab; Fx, Fy, Fzprojections of volumetric forces on the corresponding coordinate axes, referred to the unit volume; εx, εy, εz and γxy, γyz, γzx -Cauchy dependencies.…”

Section: Methodsmentioning

confidence: 99%

The Problem of the Reliability of Bending Models for Composite Plates of Medium Thickness

Shvabyuk,

Rotko,

Ribeiro

et al. 2024

JES

View full text Add to dashboard Cite

Most refined bending models of medium-thick plates, which consider transverse shear and partial compression deformations, differ little. However, despite a significant increase in the order of the governing differential equations, the results obtained from their equations give mainly a small increase in accuracy compared to the existing theories. On the other hand, such an increase in the order of the constructed systems of differential equations requires a significant increase in the effort required to solve them, complicates their physical interpretation, and narrows the range of people who can use them, primarily engineers and designers. Therefore, developing a plate-bending model that incorporates all the above factors and is on par with previously applied theories regarding the complexity of the calculation equations remains relevant. For example, most of the applied theories that do not consider transverse compression cannot be used to solve problems of contact interaction with rigid and elastic dies and bases because it is impossible to satisfy the conditions at the contact boundary of the outer surface of the plate, as well as the boundary conditions at the edges of the plate. Therefore, to provide guaranteed accuracy of the results, some researchers of these problems have introduced such a concept as “energy consistency” between the functions of representation of the displacement vector components, their number, the order of equations, and the number of boundary conditions. The authors, based on the developed version of the model of orthotropic plates of medium thickness, investigate the problem of taking into account the so-called “energy consistency” effect of the bending model, depending on the order of the design equations and the number of boundary conditions, as well as its usefulness and disadvantages in practical calculations. The equations of equilibrium in displacements and expressions for stresses in terms of force and moment forces are recorded. For rectangular and circular plates of medium thickness, test problems are solved, and the numerical data are compared with those obtained using spatial problems of elasticity theory, as well as the refined Timoshenko and Reissner theories. An analysis of the obtained results is provided.

show abstract

Section: Methodsmentioning

confidence: 99%

The Problem of the Reliability of Bending Models for Composite Plates of Medium Thickness

Shvabyuk,

Rotko,

Ribeiro

et al. 2024

JES

View full text Add to dashboard Cite

show abstract

“…Lisitsyn [86] obtained approximate analytical solutions of the three-dimensional problem of the theory of elasticity, for plates rigidly clamped along the edges using Legendre polynomials, i.e., orthogonal power polynomials. See also the work of Lisitsyn and Krivenko [87].…”

Section: Higher-order Shear Deformation Theoriesmentioning

confidence: 99%

A Review of Refined Shear Deformation Theories of Isotropic and Anisotropic Laminated Plates

Ghugal

Shimpi

2002

Journal of Reinforced Plastics and Composites

190

View full text Add to dashboard Cite

A review of displacement and stress based refined theories for isotropic and anisotropic laminated plates is presented. Various equivalent single layer and layerwise theories for laminated plates are discussed together with their merits and demerits. Exact elasticity solutions for the plate problems are cited, wherever available. Various critical issues related to plate theories are presented, based on the literature reviewed.

show abstract

“…The computations (N = 3; s = 2; 9×9) have been carried out for an isotropic plate with E = 98.1 MPa, ν = 0.25, l 1 = l 2 = 30, and 2h = 6. The deflection at the center of the plate (9×9) is equal to 0.08269 predicted by our algorithm and to 0.08593 predicted by the method of determining states [5]. The difference between results constitutes 3.8%.…”

mentioning

confidence: 95%

Numerical stress-strain analysis of a nonthin elastic plate

Chernopiskii

2006

Int Appl Mech

View full text Add to dashboard Cite

A numerical solution to elastic-equilibrium problems for nonthin plates is proposed. The solution is obtained by using the curvilinear-mesh method in combination with Vekua's method. The efficiency (rapid convergence and accuracy) of this approach is demonstrated by solving test problems for thick plates that can also be solved exactly or approximately by other methods. A numerical solution is obtained to the bending problem for orthotropic nonthin plates of constant and varying thickness Keywords: nonthin plate, constant and varying thickness, bending, numerical solutions, comparison of resultsIn searching for numerical solutions describing the elastic deformation and stability of thin plates and shells, finite-difference schemes are used to approximate the governing equations and relations of the theory of thin shells [3][4]. The discrete-orthogonalization method is successfully used to solve spatial problems for thick shells [8][9][10][11]. Note that to determine the stress-strain state (SSS) in stress concentration zones (holes or abrupt changes in the geometry of the boundary surfaces) with adequate accuracy, it is necessary to use algebraic systems of equations or stiffness matrices with a great number of variables. Such systems as a rule converge weakly when solved by iteration. The reason is that finite-difference or finite-element schemes of approximating the differential relations of the theory of shells disregard rigid-body displacements of the displacement vector.The approach proposed in [4] made it possible to efficiently solve a number of problems for thin shells in view of rigid-body displacements. It was also used to find a numerical solution to the stress-strain problem for shells and to compare it with the exact solutions of test problems.Here we set forth an efficient numerical algorithm, based on the approach from [4], for solving the elastic problem for nonthin plates.

show abstract

Calculation of clamped plates in the formation of the three-dimensional problem of the theory of elasticity

Cited by 5 publications

References 2 publications

The Problem of the Reliability of Bending Models for Composite Plates of Medium Thickness

The Problem of the Reliability of Bending Models for Composite Plates of Medium Thickness

A Review of Refined Shear Deformation Theories of Isotropic and Anisotropic Laminated Plates

Numerical stress-strain analysis of a nonthin elastic plate

Contact Info

Product

Resources

About