scholarly journals COMPARISON OF SOLUTIONS TO THE BENDING PROBLEMS OF THREE-LAYER PLATES ON THE WINKLER AND PASTERNAK FOUNDATIONS

2021 ◽  
Vol 1 (54) ◽  
pp. 30-37
Author(s):  
Anastasiya G. KOZEL ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Median Plane ◽  
Winkler Foundation ◽  
Numerical Comparison ◽  
Relative Shear ◽  
Cylindrical Coordinate ◽  
Pasternak Model ◽  
Bending Problems ◽  
Two Parameter

Solutions of problems on axisymmetric bending of an elastic three-layer circular plate on the Winkler and Pasternak foundations are given. The bearing layers are taken as isotropic, for which Kirchhoff’s hypotheses are fulfilled. In a sufficiently thick lightweight, incompressible in thickness aggregate, the Timoshenko model is valid. The cylindrical coordinate system, in which the statements and solutions of boundary value problems are carried out, is connected with the median plane of the filler. On the plate contour, it is assumed that there is a rigid diaphragm that prevents the relative shear of the layers. The system of differential equations of equilibrium is obtained by the variational method. Three types of boundary conditions are formulated. One- and two-parameter Winkler and Pasternak models are used to describe the reaction of an elastic foundation. The solution to the boundary value problem is reduced to finding three desired functions, plate deflection, shear, and radial displacement in the filler. The general analytical solution to the boundary value problem is written out in the case of the Pasternak model in Bessel functions. At the Winkler foundation, the known solution is given in Kelvin functions. A numerical comparison of the displacements and stresses obtained by both models with a uniformly distributed load and rigid sealing of the plate contour is carried out.

scholarly journals Bending by concentrated force of thin plate located on elastic foundation and weakened by contact crack

2019 ◽  
pp. 118-121
Author(s):  
M. V. Makoviichuk ◽  
I. P. Shatskyi
Keyword(s):  
Intensity Factor ◽  
Boundary Value Problem ◽  
Elastic Foundation ◽  
Crack Closure ◽  
Boundary Value ◽  
Limit Load ◽  
Winkler Foundation ◽  
Intensity Factors ◽  
Concentrated Force ◽  
Crack Line

The paper considers the two-dimensional formulation of the problem of the contact interaction of the crack edges in a plate bent by the concentrated force on the elastic Winkler foundation. The crack closure is described using the model of contact along a line in one of the plate surfaces. Within the framework of this model, the boundary value problem is formulated for the equations of the classical theories of plate bending on the elastic foundation and a plane stress state with interrelated tension and bending conditions on the crack line. The obtained boundary value problem has been solved using singular integral equations method. Based on numerical solutions of the integral equation the dependences of forces and moments intensity factors in the vicinity of the defect tips and distribution of contact forces along the crack line on the parameters of elastic foundation stiffness and the coordinate of the application point of the load have been investigated. The effect of crack closure and influence of the elastic foundation stiffness on the limit equilibrium of the plate, depending on the coordinate of the point of application of the concentrated force, has been evaluated. The area of the correctness of the problem statement when the crack closure occurs throughout its length has been established. It was found that the crack closure leads to the appearance of nonzero forces intensity factor, reduction of the moments intensity factor and increase of the limit load. The dependences of the forces and moments intensity factors and the limit load on the dimensionless coordinate of the point of application of the concentrated force are nonmonotonic. Numerical analysis showed that increasing the elastic foundation stiffness, as well as the displacement of the point of application of the force from the center of the cut, increase the limit load and weaken the contact reaction.

scholarly journals ELASTIC BENDING OF A THREE-LAYER CIRCULAR PLATE WITH STEP-VARIABLE THICKNESS

2021 ◽  
Vol 1 (54) ◽  
pp. 25-29
Author(s):  
Denis V. LEONENKO ◽  
Keyword(s):  
Boundary Value Problem ◽  
Circular Plate ◽  
Variable Thickness ◽  
Boundary Value ◽  
Median Plane ◽  
Inhomogeneous System ◽  
Equilibrium Equations ◽  
Elastic Circular Plate ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

The bending of a three-layer elastic circular plate with step-variable thickness is considered. To describe kinematics of asymmetrical in thickness core pack, the broken line hypotheses are accepted. In thin bearing layers, Kirchhoff’s hypotheses are valid. In a relatively thick filler incompressible in thickness, Timoshenko’s hypothesis on the straightness and incompressibility of the deformed normal is fulfilled. The formulation of the corresponding boundary value problem is presented. Equilibrium equations are obtained by the variational Lagrange method. The solution of the boundary value problem is reduced to finding three required functions in each section, deflection, shear and radial displacement of the median plane of the filler. An inhomogeneous system of ordinary linear differential equations is obtained for these functions. The boundary conditions correspond to rigid pinching of the plate contour. A parametric analysis of the obtained solution is carried out.

scholarly journals Efficient approximate analytical methods for nonlinear fuzzy boundary value problem

2022 ◽  
Vol 12 (2) ◽  
pp. 1916
Author(s):  
Ali Fareed Jameel ◽  
Hafed H Saleh ◽  
Amirah Azmi ◽  
Abedel-Karrem Alomari ◽  
Nidal Ratib Anakira ◽  
...  
Keyword(s):  
Boundary Value Problem ◽  
Analytical Methods ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Numerical Comparison ◽  
Fuzzy Boundary Value Problems ◽  
Fuzzy Boundary ◽  
Approximate Analytical Methods

This paper aims to solve the nonlinear two-point fuzzy boundary value problem (TPFBVP) using approximate analytical methods. Most fuzzy boundary value problems cannot be solved exactly or analytically. Even if the analytical solutions exist, they may be challenging to evaluate. Therefore, approximate analytical methods may be necessary to consider the solution. Hence, there is a need to formulate new, efficient, more accurate techniques. This is the focus of this study: two approximate analytical methods-homotopy perturbation method (HPM) and the variational iteration method (VIM) is proposed. Fuzzy set theory properties are presented to formulate these methods from crisp domain to fuzzy domain to find approximate solutions of nonlinear TPFBVP. The presented algorithms can express the solution as a convergent series form. A numerical comparison of the mean errors is made between the HPM and VIM. The results show that these methods are reliable and robust. However, the comparison reveals that VIM convergence is quicker and offers a swifter approach over HPM. Hence, VIM is considered a more efficient approach for nonlinear TPFBVPs.

scholarly journals Existence of Solutions of a Discrete Fourth-Order Boundary Value Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Ruyun Ma ◽  
Chenghua Gao ◽  
Yongkui Chang
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Linear Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Linear Eigenvalue Problem ◽  
Existence And Multiplicity ◽  
Nonresonance Condition ◽  
Two Parameter

Leta,bbe two integers withb-a≥5and let𝕋2={a+2,a+3,…,b-2}. We show the existence of solutions for nonlinear fourth-order discrete boundary value problemΔ4u(t-2)=f(t,u(t),Δ2u(t-1)),t∈𝕋2,u(a+1)=u(b-1)=Δ2u(a)=Δ2u(b-2)=0under a nonresonance condition involving two-parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem.

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