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Hybrid stress analysis using digitized photoelastic data and numerical methods
 Date Issued:
 1989
 Summary:
 Equations of stressdifference elasticity, derived from the equations of equilibrium and compatibility for a twodimensional stress field, are solved for arbitrarily digitized, singly and multiply connected domains. Photoelastic data determined experimentally along the boundary provide the boundary values for the solution of the three elliptic partial differential equations by the finite difference method. A computerized method is developed to generate grid mesh, weighting functions and nodal connectivity within the digitized boundary for the solution of these partial differential equations. A method is introduced to digitize the photoelastic fringes, namely isochromatics and isoclinics, and to estimate the values of sigma1  sigma2, sigma x  sigma y and tau xy at each nodal point by an interpolation technique. Interpolated values of the stress parameters are used to improve the initial estimate and hence the convergence of the iterative solution of the system of equations. Superfluous boundary conditions are added from the digitized photoelastic data for further speeding up the rate of convergence. The boundary of the domain and the photoelastic fringes are digitized by physically traversing the cursor along the boundary, and the digitized information is scanned horizontally and vertically to generate internal and boundary nodal points. A linear search determines the nodal connectivity and isolates the boundary points for the input of the boundary values. A similar scanning method estimates the photoelastic parameters at each nodal point and also finds the points closest to the tint of passage of each photoelastic fringe. Stress values at these close points are determined without interpolation and are subsequently used as superfluous boundary conditions in the iteration scheme. Successive overrelaxation is applied to the classical GaussSeidel method for final enhancement of the convergence of the iteration process. The iteration scheme starts with an accelerating factor other than unity and estimates the spectral radius of the iteration matrix from the two vector norms. This information is used to estimate a temporary value of the optimum relaxation parameter, omega[opt], which is used for a fixed number of iterations to approximate a better value of the accelerating factor. The process is continued until two successive estimates differ by a given tolerance or the stopping criteria are reached. Detailed techniques of developing the code for mesh generation, photoelastic data collection and boundary value interpolation to solve the elliptic boundary value problems are presented. Three separate examples with varying stress gradients and fringe patterns are presented to test the validity of the code and the overall method. Results are compared with the analytical and experimental solutions, and the significant improvement in the rate of convergence is demonstrated.
Title:  Hybrid stress analysis using digitized photoelastic data and numerical methods. 
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Name(s): 
Mahfuz, Hassan Florida Atlantic University, Degree grantor Case, Robert O., Thesis advisor College of Engineering and Computer Science Department of Ocean and Mechanical Engineering 

Type of Resource:  text  
Genre:  Electronic Thesis Or Dissertation  
Issuance:  monographic  
Date Issued:  1989  
Publisher:  Florida Atlantic University  
Place of Publication:  Boca Raton, Fla.  
Physical Form:  application/pdf  
Extent:  167 p.  
Language(s):  English  
Summary:  Equations of stressdifference elasticity, derived from the equations of equilibrium and compatibility for a twodimensional stress field, are solved for arbitrarily digitized, singly and multiply connected domains. Photoelastic data determined experimentally along the boundary provide the boundary values for the solution of the three elliptic partial differential equations by the finite difference method. A computerized method is developed to generate grid mesh, weighting functions and nodal connectivity within the digitized boundary for the solution of these partial differential equations. A method is introduced to digitize the photoelastic fringes, namely isochromatics and isoclinics, and to estimate the values of sigma1  sigma2, sigma x  sigma y and tau xy at each nodal point by an interpolation technique. Interpolated values of the stress parameters are used to improve the initial estimate and hence the convergence of the iterative solution of the system of equations. Superfluous boundary conditions are added from the digitized photoelastic data for further speeding up the rate of convergence. The boundary of the domain and the photoelastic fringes are digitized by physically traversing the cursor along the boundary, and the digitized information is scanned horizontally and vertically to generate internal and boundary nodal points. A linear search determines the nodal connectivity and isolates the boundary points for the input of the boundary values. A similar scanning method estimates the photoelastic parameters at each nodal point and also finds the points closest to the tint of passage of each photoelastic fringe. Stress values at these close points are determined without interpolation and are subsequently used as superfluous boundary conditions in the iteration scheme. Successive overrelaxation is applied to the classical GaussSeidel method for final enhancement of the convergence of the iteration process. The iteration scheme starts with an accelerating factor other than unity and estimates the spectral radius of the iteration matrix from the two vector norms. This information is used to estimate a temporary value of the optimum relaxation parameter, omega[opt], which is used for a fixed number of iterations to approximate a better value of the accelerating factor. The process is continued until two successive estimates differ by a given tolerance or the stopping criteria are reached. Detailed techniques of developing the code for mesh generation, photoelastic data collection and boundary value interpolation to solve the elliptic boundary value problems are presented. Three separate examples with varying stress gradients and fringe patterns are presented to test the validity of the code and the overall method. Results are compared with the analytical and experimental solutions, and the significant improvement in the rate of convergence is demonstrated.  
Identifier:  11934 (digitool), FADT11934 (IID), fau:8853 (fedora)  
Collection:  FAU Electronic Theses and Dissertations Collection  
Note(s): 
College of Engineering and Computer Science Thesis (Ph.D.)Florida Atlantic University, 1989. 

Subject(s): 
Strains and stresses Photoelasticity Numerical analysisData processing 

Held by:  Florida Atlantic University Libraries  
Persistent Link to This Record:  http://purl.flvc.org/fcla/dt/11934  
Sublocation:  Digital Library  
Use and Reproduction:  Copyright © is held by the author, with permission granted to Florida Atlantic University to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.  
Use and Reproduction:  http://rightsstatements.org/vocab/InC/1.0/  
Host Institution:  FAU  
Is Part of Series:  Florida Atlantic University Digital Library Collections. 