Exact Elasticity Solution for the Density Functionally Gradient Beam by Using Airy Stress Function

2011 ◽  
Vol 110-116 ◽  
pp. 4669-4676 ◽  
Author(s):  
Alireza R. Daneshmehr ◽  
Saeed Momeni ◽  
Mahdi Reza Akhloumadi
Keyword(s):  
Boundary Conditions ◽  
Stress Function ◽  
Rectangular Cross Section ◽  
Airy Stress Function ◽  
Uniform Load ◽  
Functionally Gradient ◽  
Different Types ◽  
Classical Boundary ◽  
Clamped Beams ◽  
Fixed End

In this paper the problem of a density-functionally gradient beam subjected to uniform load is studied. Airy stress function methodology is used to obtain a set of analytical solutions for simply supported and clamped beams subjected to uniform load. A stress function in the form of polynomial is proposed and determined. The treatment for fixed-end boundary conditions is the same as that presented by Timoshenko and Goodier (1970). By this method all of the analytical plane-stress solutions can be obtained for a uniformly loaded isotropic beam with rectangular cross section under different types of classical boundary conditions.

Stress Function Interface and Boundary Conditions in Anisotropic Materials

1982 ◽  
Vol 49 (4) ◽  
pp. 787-791 ◽  
Author(s):  
E. E. Gdoutos ◽  
M. Kattis
Keyword(s):  
Boundary Conditions ◽  
Stress Function ◽  
Anisotropic Media ◽  
Anisotropic Materials ◽  
Function Approach ◽  
Airy Stress Function ◽  
Stress Functions ◽  
The Media

The stress and displacement continuity conditions for interfaces between two different anisotropic media were formulated in terms of the Airy stress functions of the media. It was shown that such formulation greatly facilitates the solution of the problems of composite anisotropic materials by the Airy stress function approach. Two examples were given to demonstrate the potentiality of the method.

Simplified Approach to Solution of Mixed Boundary Value Problems on Homogeneous Circular Domain in Elasticity

2018 ◽  
Vol 86 (2) ◽  
Author(s):  
Gaurav Singh ◽  
Tanmay K. Bhandakkar
Keyword(s):  
Boundary Conditions ◽  
Stress Function ◽  
Mixed Boundary ◽  
Linear Equations ◽  
Mixed Boundary Conditions ◽  
Circular Domain ◽  
Infinite Plane ◽  
Airy Stress Function ◽  
System Of Linear Equations ◽  
Starting Point

This work proposes a novel strategy to render mixed boundary conditions on circular linear elastic homogeneous domain to displacement-based condition all along the surface. With Michell solution as the starting point, the boundary conditions and extent of the domain are used to associate the appropriate type and number of terms in the Airy stress function. Using the orthogonality of sine and cosine functions, the modified boundary conditions lead to a system of linear equations for the unknown coefficients in the Airy stress function. Solution of the system of linear equations provides the Airy stress function and subsequently stresses and displacement. The effectiveness of the present approach in terms of ease of implementation, accuracy, and versatility to model variants of circular domain is demonstrated through excellent comparison of the solution of following problems: (i) annulus with mixed boundary conditions on outer radius and prescribed traction on the inner radius, (ii) cavity surface with mixed boundary conditions in an infinite plane subjected to far-field uniaxial loading, and (iii) circular disc constrained on part of the surface and subjected to uniform pressure on rest of the surface.

scholarly journals DIAGRAMS FOR STRESS AND DEFLECTION PREDICTION IN CROSS-LAMINATED TIMBER (CLT) PANELS WITH NON-CLASSICAL BOUNDARY CONDITIONS

2020 ◽  
Vol 14 (1) ◽  
Author(s):  
Nikola Obradović ◽  
Marija Todorović ◽  
Miroslav Marjanović ◽  
Emilija Damnjanović
Keyword(s):  
Boundary Conditions ◽  
Engineering Practice ◽  
Design Codes ◽  
Detailed Design ◽  
Cross Laminated Timber ◽  
Simply Supported ◽  
Laminar Composites ◽  
Different Types ◽  
Practical Engineering ◽  
Classical Boundary

Invention of cross-laminated timber (CLT) was a big milestone for building with wood. Due tonovelty of CLT and timber’s complex mechanical behavior, the existing design codes cover onlyrectangular CLT panels, simply supported along 2 parallel or all 4 edges, making numerical methodsnecessary in other cases. This paper presents a practical engineering tool for stress and deflectionprediction of CLT panels with non-classical boundary conditions, based on the software for thecomputational analysis of laminar composites, previously developed by the authors. Diagramsapplicable in engineering practice are developed for some common cases. The presentedmethodology could be a basis for more detailed design handbooks and guidelines for various layoutsof CLT panels and different types of loadings.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

scholarly journals Flexural-Torsional Vibration of Thin-Walled Beams Subjected to Combined Initial Axial Load and End Bending Moment: Application to the Design of Saw Tooth Blades

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Van Binh Phung ◽  
Anh Tuan Nguyen ◽  
Hoang Minh Dang ◽  
Thanh-Phong Dao ◽  
V. N. Duc
Keyword(s):  
Boundary Conditions ◽  
Axial Load ◽  
Stability Boundary ◽  
Bending Moment ◽  
Element Analysis ◽  
The Finite Element Analysis ◽  
Rayleigh Method ◽  
Thin Walled ◽  
The Stability ◽  
Fixed End

The present paper analyzes the vibration issue of thin-walled beams under combined initial axial load and end moment in two cases with different boundary conditions, specifically the simply supported-end and the laterally fixed-end boundary conditions. The analytical expressions for the first natural frequencies of thin-walled beams were derived by two methods that are a method based on the existence of the roots theorem of differential equation systems and the Rayleigh method. In particular, the stability boundary of a beam can be determined directly from its first natural frequency expression. The analytical results are in good agreement with those from the finite element analysis software ANSYS Mechanical APDL. The research results obtained here are useful for those creating tooth blade designs of innovative frame saw machines.

scholarly journals An Advanced Galerkin Approach to Solve the Nonlinear \\[6pt]Reaction-Diffusion Equations With Different Boundary Conditions

2022 ◽  
Vol 14 (1) ◽  
pp. 30
Author(s):  
Hazrat Ali ◽  
Md. Kamrujjaman ◽  
Md. Shafiqul Islam
Keyword(s):  
Boundary Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Parabolic Partial Differential Equations ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Nonlinear Parabolic ◽  
Comparison Results ◽  
Different Types ◽  
Convergence And Stability Analysis

This study proposed a scheme originated from the Galerkin finite element method (GFEM) for solving nonlinear parabolic partial differential equations (PDEs) numerically with initial and different types of boundary conditions. The scheme is applied generally handling the nonlinear terms in a simple way and throwing over restrictive assumptions. The convergence and stability analysis of the method are derived. The error of the method is estimated. In the series, eminent problems are solved, such as  Fisher's equation, Newell-Whitehead-Segel equation, Burger's equation, and  Burgers-Huxley equation to demonstrate the validity, efficiency, accuracy, simplicity and applicability of this scheme. In each example, the comparison results are presented both numerically and graphically

scholarly journals Form-Finding of Shells Containing Both Tension and Compression Using the Airy Stress Function

2020 ◽  
Author(s):  
Masaaki Miki ◽  
Emil Adiels ◽  
William Baker ◽  
Toby Mitchell ◽  
Alexander Sehlstrom ◽  
...  
Keyword(s):  
Stress Function ◽  
Mixed Type ◽  
Airy Stress Function ◽  
Form Finding ◽  
Graphic Statics ◽  
Tension And Compression ◽  
Stress Functions ◽  
Complete Set ◽  
Pure Compression ◽  
Asymptotic Lines

Pure-compression shells have been the central topic in the form-finding of shells. This paper studies tension-compression mixed type shells by utilizing a NURBS-based isogeometric form-finding approach that analyzes Airy stress functions to expand the possible plan geometry. A complete set of smooth version graphic statics tools is provided to support the analyses. The method is validated using examples with known solutions, and a further example demonstrates the possible forms of shells that the proposed method permits. Additionally, a guideline to configure a proper set of boundary conditions is presented through the lens of asymptotic lines of the stress functions.

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