Stress Analysis of Cracked Rectangular Beams

1967 ◽  
Vol 34 (3) ◽  
pp. 693-701 ◽  
Author(s):  
R. A. Westmann ◽  
W. H. Yang
Keyword(s):  
Stress Analysis ◽  
Boundary Value Problems ◽  
Cross Section ◽  
Mixed Boundary ◽  
Rectangular Cross Section ◽  
Boundary Value ◽  
Longitudinal Shear ◽  
Physical Interest ◽  
Dual Series Equations ◽  
Edge Cracks

This paper is concerned with the solution of mixed boundary-value problems arising from the longitudinal shear and Saint-Venant torsion and flexure of cracked prismatic beams. The members considered have a rectangular cross section with planar edge cracks extending the length of the beam. The problem is reduced to a pair of dual-series equations, the solution of which is obtained by a method due to Sneddon and Srivastav [1]. Numerical results are presented for a variety of quantities of physical interest.

scholarly journals On Triple Series Equations Involving Series of Jacobi Polynomials

1967 ◽  
Vol 15 (3) ◽  
pp. 221-231 ◽  
Author(s):  
K. N. Srivastava
Keyword(s):  
Boundary Value Problems ◽  
Legendre Polynomials ◽  
Jacobi Polynomials ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Value Problems ◽  
Dual Series Equations

Recently Collins (2) has studied triple series equations involving series of Legendre polynomials. These equations arise in the study of mixed boundary value problems and can be regarded as extensions of the dual series equations considered by Collins in (1).

Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Roland Duduchava
Keyword(s):  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Beltrami Equation ◽  
Bessel Potential ◽  
Convolution Operators ◽  
Classical Setting ◽  
Type Boundary ◽  
Mellin Convolution ◽  
Bessel Potential Spaces

AbstractThe purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 in the non-classical setting when solutions are sought in the Bessel potential spaces \mathbb{H}^{s}_{p}(\mathcal{C}), \frac{1}{p}<s<1+\frac{1}{p}, 1<p<\infty. Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain \Omega_{\alpha}\subset\mathbb{R}^{2} of magnitude 𝛼. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J.27 (2020), 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2\times 2 systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes \mathbb{R}^{+} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys.60 (2013), 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl.443 (2016), 2, 707–731].

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