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A closed form solution for stress concentration around a circular hole in a lIAEME Publication
This document presents a closed-form solution for determining stress concentration around a circular hole in an infinite plate with linearly varying stress. The plate is subjected to a tensile stress that varies linearly from the top edge to the bottom edge. An equation is derived for the stress field in polar coordinates using stress functions. Boundary conditions are applied at the hole edge and at a large distance from the hole. A solution is obtained for the stresses around the hole in terms of constants and the original varying stress field, without requiring numerical methods. The results are compared to finite element analysis and show close agreement.
This document discusses plate and shell elements for structural analysis. Plate elements are used to model flat surfaces, while shell elements model curved surfaces. Kirchhoff plate theory and Reissner-Mindlin plate theory are described for modeling plate bending, with the latter including transverse shear deformations. The derivation of a rectangular plate bending element is shown, involving assumed displacement fields and strain-curvature relationships. Shell elements can be formulated by combining plate and plane stress elements. Limitations of Kirchhoff shell elements for nearly coplanar or folded plate structures are noted.
This document provides an introduction to the theory of plates, which are structural elements that are thin and flat. It defines what is meant by a thin plate and discusses different plate classifications based on thickness. The document derives the basic equations that describe plate behavior by taking advantage of the plate's thin, planar character. It also discusses three-dimensional considerations like stress components, equilibrium, strain and displacement for putting the plate theory into context.
This document analyzes the vibration of a square plate with a circular cutout using finite element analysis. It studies how the natural frequency of the plate is affected by the diameter of the circular hole. The plate is modeled in ANSYS using solid elements with simply supported and clamped-free boundary conditions. Results show the natural frequency decreases with increasing hole diameter, with a more significant effect at higher modes. Mode shapes are also generated. Parametric studies are performed to analyze the relationship between hole diameter and natural frequency. The results are verified with data from previous studies.
Application of CAD and SLA Method in Dental ProsthesisIDES Editor
Placement of dental implants requires precise
planning that accounts for anatomic limitations and
restorative goals. Diagnosis can be made with the assistance
of computerized tomographic scanning, but transfer of
planning to the surgical field is limited. Precise implant
placement no longer relies upon so called mental-navigation
but rather can be computer guided, based on a three
dimensional prosthetically directed plan. Recently, novel CAD/
CAM techniques such as stereolithographic rapid prototyping
have been developed to build surgical guides in an attempt to
improve precision of implant placement. The purpose of this
paper is to discuss the use of scanning equipments to transfer
clinically relevant prosthetic information which can be used
for fabrication of stereolithographic medical models and
surgical guides. The proposed method provides solid evidence
that computer-aided design and manufacturing technologies
may become a new avenue for custom-made dental implants
design, analysis, and production in the 21st century.
techDynamic characteristics and stability of cylindrical textured journal bea...ijmech
This document summarizes a study on the dynamic characteristics and stability of cylindrical textured journal bearings. The researchers numerically solved the Reynolds equation to analyze the effect of texture depth and density on the stiffness, damping coefficients, stability parameter, and whirl ratio of the bearing. They found that stability is enhanced with increasing texture depth, while there is an optimum texture density that results in maximum stability. Direct and cross stiffness/damping coefficients, stability margin, and whirl ratio were presented for different texture parameters.
This document describes an aerospace engineering final project analyzing the prediction of laminate deformation using an analytical method. Three different stacking sequences of a laminated composite plate are analyzed: symmetric and balanced, symmetric and unbalanced, and asymmetric and balanced. The classical lamination theory and derivations of the deformation equations are presented. Results from MATLAB calculations are compared to ANSYS finite element analysis results for the different stacking sequences and loading conditions, showing good agreement with errors generally under 3%.
This document provides an overview of basic equations for the theory of plates and shells. It discusses the state of stress and strain at a point, including defining the six independent stress and strain components. It presents the relationships between strain and displacement, and discusses the equilibrium equations relating stress and body forces. Finally, it provides the equations for both Cartesian and cylindrical coordinate systems. The key concepts covered are the fundamental equations that form the basis of plate and shell theory.
A closed form solution for stress concentration around a circular hole in a lIAEME Publication
This document presents a closed-form solution for determining stress concentration around a circular hole in an infinite plate with linearly varying stress. The plate is subjected to a tensile stress that varies linearly from the top edge to the bottom edge. An equation is derived for the stress field in polar coordinates using stress functions. Boundary conditions are applied at the hole edge and at a large distance from the hole. A solution is obtained for the stresses around the hole in terms of constants and the original varying stress field, without requiring numerical methods. The results are compared to finite element analysis and show close agreement.
This document discusses plate and shell elements for structural analysis. Plate elements are used to model flat surfaces, while shell elements model curved surfaces. Kirchhoff plate theory and Reissner-Mindlin plate theory are described for modeling plate bending, with the latter including transverse shear deformations. The derivation of a rectangular plate bending element is shown, involving assumed displacement fields and strain-curvature relationships. Shell elements can be formulated by combining plate and plane stress elements. Limitations of Kirchhoff shell elements for nearly coplanar or folded plate structures are noted.
This document provides an introduction to the theory of plates, which are structural elements that are thin and flat. It defines what is meant by a thin plate and discusses different plate classifications based on thickness. The document derives the basic equations that describe plate behavior by taking advantage of the plate's thin, planar character. It also discusses three-dimensional considerations like stress components, equilibrium, strain and displacement for putting the plate theory into context.
This document analyzes the vibration of a square plate with a circular cutout using finite element analysis. It studies how the natural frequency of the plate is affected by the diameter of the circular hole. The plate is modeled in ANSYS using solid elements with simply supported and clamped-free boundary conditions. Results show the natural frequency decreases with increasing hole diameter, with a more significant effect at higher modes. Mode shapes are also generated. Parametric studies are performed to analyze the relationship between hole diameter and natural frequency. The results are verified with data from previous studies.
Application of CAD and SLA Method in Dental ProsthesisIDES Editor
Placement of dental implants requires precise
planning that accounts for anatomic limitations and
restorative goals. Diagnosis can be made with the assistance
of computerized tomographic scanning, but transfer of
planning to the surgical field is limited. Precise implant
placement no longer relies upon so called mental-navigation
but rather can be computer guided, based on a three
dimensional prosthetically directed plan. Recently, novel CAD/
CAM techniques such as stereolithographic rapid prototyping
have been developed to build surgical guides in an attempt to
improve precision of implant placement. The purpose of this
paper is to discuss the use of scanning equipments to transfer
clinically relevant prosthetic information which can be used
for fabrication of stereolithographic medical models and
surgical guides. The proposed method provides solid evidence
that computer-aided design and manufacturing technologies
may become a new avenue for custom-made dental implants
design, analysis, and production in the 21st century.
techDynamic characteristics and stability of cylindrical textured journal bea...ijmech
This document summarizes a study on the dynamic characteristics and stability of cylindrical textured journal bearings. The researchers numerically solved the Reynolds equation to analyze the effect of texture depth and density on the stiffness, damping coefficients, stability parameter, and whirl ratio of the bearing. They found that stability is enhanced with increasing texture depth, while there is an optimum texture density that results in maximum stability. Direct and cross stiffness/damping coefficients, stability margin, and whirl ratio were presented for different texture parameters.
This document describes an aerospace engineering final project analyzing the prediction of laminate deformation using an analytical method. Three different stacking sequences of a laminated composite plate are analyzed: symmetric and balanced, symmetric and unbalanced, and asymmetric and balanced. The classical lamination theory and derivations of the deformation equations are presented. Results from MATLAB calculations are compared to ANSYS finite element analysis results for the different stacking sequences and loading conditions, showing good agreement with errors generally under 3%.
This document provides an overview of basic equations for the theory of plates and shells. It discusses the state of stress and strain at a point, including defining the six independent stress and strain components. It presents the relationships between strain and displacement, and discusses the equilibrium equations relating stress and body forces. Finally, it provides the equations for both Cartesian and cylindrical coordinate systems. The key concepts covered are the fundamental equations that form the basis of plate and shell theory.
Design and analysis of laminated composite plates using MATLABkartik kulkarni
This document discusses the analysis of laminated composite plates using finite element analysis in ANSYS and classical lamination theory calculations in MATLAB. It analyzes simply supported symmetrically laminated composite plates subjected to uniform loads. First, a 4-ply composite plate is modeled and analyzed in ANSYS, and the results are validated with MATLAB calculations. Different fiber orientations are then investigated. Next, a 20-ply plate is modeled and analyzed, revealing that bending stiffness values D16 and D26 become insignificant for thicknesses over 16 plies. The document concludes that fiber orientation, stacking sequence, and number of plies all affect deflection results.
This document presents the development of a state space equation approach to obtain the three-dimensional solution for thick orthotropic plates with symmetric clamped-free edges. The state space method allows for an exact solution that satisfies all equations of elasticity and accounts for all elastic constants. The system matrix, which is a key part of the state space solution, is derived for symmetric clamped-free boundary conditions. Modal expansions are used to develop sixth-order differential equations governing the transverse displacement, whose solutions provide the stresses and displacements across the plate thickness.
Kantorovich-Vlasov Method for Simply Supported Rectangular Plates under Unifo...IJCMESJOURNAL
In this study, the Kantorovich-Vlasov method has been applied to the flexural analysis of simply supported Kirchhoff plates under transverse uniformly distributed load on the entire plate domain. Vlasov method was used to construct the coordinate functions in the x direction and the Kantorovich method was used to consider the assumed displacement field over the plate. The total potential energy functional and the corresponding Euler-Lagrange equations were obtained. This was solved subject to the boundary conditions to obtain the displacement field over the plate. Bending moments were then obtained using the moment curvature equations. The solutions obtained were rapidly convergent series for deflection, and bending moments. Maximum deflection and maximum bending moments occurred at the center and were also obtained as rapidly convergent series. The series were computed for varying plate aspect ratios. The results were identical with Levy-Nadai solutions for the same problem.
Three dimensional static analysis of two dimensional functionally graded platesijmech
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated. The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it shows very good agreement.
Three dimensional static analysis of two dimensional functionally graded platesrtme
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional
theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The
effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated.
The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it
shows very good agreement
This summary analyzes the free vibration of laminated composite and sandwich plates using the Euler-Lagrange equation based on first order shear deformation theory. The document presents analytical formulations and solutions for the natural frequency of simply supported composite and sandwich plates. The results are compared to previous literature. The theoretical model accounts for transverse shear deformation, transverse normal strain/stress, and nonlinear variation of in-plane displacement through the thickness, modeling warping more accurately without shear correction coefficients.
A COMPARATIVE STUDY OF VARIOUS METHODS TO EVALUATE IMPEDANCE FUNCTION FOR SHA...Samirsinh Parmar
Impedance function, Foundation Vibration, dynamic soil-structure interaction, Barkan, Dominguez, Dobry and Gazetas for evaluation of impedance functions for various modes of vibration of shallow foundation
This document presents a higher-order finite element model to predict the linear buckling behavior of laminated beam structures. The model is based on a displacement field that assumes a non-linear variation of displacements through the beam width and depth, eliminating the need for shear correction factors. Numerical applications show the model accurately predicts buckling loads for isotropic and anisotropic beams of various cross-sections when compared to other models and closed-form solutions. The higher-order model is more versatile and accurate than lower-order models like Euler-Bernoulli for a range of slenderness ratios and beam geometries.
This document analyzes the deflections and stresses in composite laminated plates using the dynamic relaxation method. It summarizes the following key points:
1) The dynamic relaxation method is used to solve the equilibrium equations for laminated plate deformation. This method converts the equations to dynamic form by adding damping and inertia terms.
2) Factors like shear deformation, length-to-thickness ratio, number of layers, material anisotropy, and fiber orientation affect plate deflection based on the analysis of square laminated plates with uniform loading.
3) Linear analysis overpredicts deflection compared to the dynamic relaxation method, which accounts for large deflections. Deflection depends on load size, ply orientation, and other
IRJET- Comparison Between Thin Plate and Thick Plate from Navier Solution us...IRJET Journal
This document compares thin plate theory and thick plate theory using MATLAB software. It derives expressions for bending moments, twisting moments, and shear forces for both Kirchhoff's thin plate theory and Reissner's thick plate theory. Detailed programs are written in MATLAB to analyze rectangular plates of varying thickness under Navier's solution. The results show that stress resultants from thin plate theory are independent of thickness, while thick plate theory accounts for thickness in its expressions. Plots demonstrate how bending moments, twisting moments, and shear forces vary across plates of different thicknesses according to each theory.
The document discusses analytical expressions for notch stress intensity factors (NSIFs) under torsion loading for circumferentially-notched rounded bars. It begins by reviewing previous work on NSIFs and stress fields near notches and cracks. It then presents an exact, closed-form solution for the mode III NSIF of a deep V-notch and extends this solution to finite notches using a shape function. The goal is to determine NSIFs directly from theoretical stress concentration factors in order to better understand notch mechanics.
Reliability Analysis of the Sectional Beams Due To Distribution of Shearing S...researchinventy
This paper shows the results of the Reliability Analysis of the sectional beams due to distribution of Shear Stress. It is assumed that the load was uniformly distributed over the beam. It is discussed that the distribution of shear stress over the beam. It is discussed that the average shears stress and maximum shear stress across the section of the beam for Weibull distribution. The reliability analysis of distribution of shearing stresses over sectional beams is performed. Also it is derived that the hazard functions for these types of beams. Reliability comparison has also been done for the sectional beams. It is observed that the reliability of the beam decreased when the width (b) of the beam decreases, and the load (F) is high. The reliability of the beam is increased when the height (h) of the triangular section increases , diameter(d) of the circular beam is increased and parameter 푘 decreasses
Analysis of stiffened plate using FE ApproachIJMER
The objective of the present investigation is to study the strengthening effect of the stiffeners
on the buckling of unperforated and perforated plate when they are reinforced in longitudinal and
transverse directions. The plate is subjected to inplane uniform uniaxial end compression load having
simply supported plate boundary condition. The parameters considered are plate aspect ratio, area ratio and types of stiffeners. The analysis has been carried out using ANSYS finite element software. The buckling analysis shows that the influence of transverse stiffener is less when compared to longitudinal stiffener
An axisymmetric bending and shear stress analysis of of functionally graded ...eSAT Journals
Abstract The effects of exposure from electromagnetic radiations of wireless cellular transmission towers to human health have attracted the attention of many researchers. Different works have revealed the harmful of electromagnetic radiation exposure to human health based on distance from the source and period of exposure. As one stays closer and at a pro-longed period from the transmission sites, the possibility of being affected by the radiation source becomes higher. In this work, we review some of the works on assessment of electromagnetic radiation exposure and propose measures for determining safety zones based on the cases of cellular transmission towers in the Tanzania environment to avoid extended exposure to electromagnetic radiation. Key words- Cellular transmission towers; Electromagnetic radiations; Health effects; Exposure limits
APPLICATION OF THE BOUNDARY ELEMENT METHOD IN DETERMINING THE CRITICAL BUCKLI...Engenheiro Civil
The boundary element method (BEM) is used in this work to obtain the critical buckling parameters of
perforated plates. The plates have square geometry with central square hole. Compression is unidirectional and
uniformly applied at opposite edges. The values for buckling parameters are obtained for many values of thickness.
The shear deformation effect is included in the bending of plates using the isotropic model. The effect of geometric
nonlinearity is introduced by adding two integral in the formulation of the BEM: one is applied in the domain and the
other in the contour. The boundary integral can be related to one of the natural conditions according to the boundary
value problem. Quadratic continuous and discontinuous boundary elements were used. The source points were
positioned on the boundary. The singularity subtraction technique and the transformation of variables were used for
the Cauchy and weak type of singularities, respectively, when the integration is performed in elements containing the
source point. Rectangular cells were used to discretize the domain integral related to the geometric nonlinearity effect.
The results were compared with other authors.
This document provides an overview of plate bending theory. It discusses how plate theory models bending in thin plates using Kirchhoff's plate theory and thick plates using Mindlin plate theory. Kirchhoff's plate theory assumes plates bend without shear deformation and strains are related to plate deflection. Mindlin plate theory allows for shear deformation and separates plate rotation and deflection. The document also discusses deriving plate stresses and forces, applying boundary conditions, and using triangular plate bending elements in finite element analysis.
This document summarizes key concepts in plate theory from Chapter 13. Some key points:
- Flat plates are structural members where the thickness is small relative to length and width, with the middle surface lying in a plane. Plate theory reduces the 3D problem to 2D.
- Stress resultants in plates include in-plane forces (Nxx, Nyy, Nxy), bending moments (Mxx, Myy), and shears (Qx, Qy). Boundary conditions for plates include no displacement, no slope, no moment, and no shear.
- Closed-form solutions exist for rectangular and circular plates with various edge conditions (simply supported, fixed) subjected to uniform loads
International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document presents a third order shear deformation theory to analyze flexure of thick cantilever beams. The theory uses a sinusoidal function in the displacement field to account for transverse shear deformation effects through the beam thickness. Governing equations and boundary conditions are derived using the principle of virtual work. Numerical examples of a cantilever beam with a cosine load distribution are presented and displacement, stress results are obtained in non-dimensional form. The results are discussed and compared to other beam theories to demonstrate the efficiency of the third order shear deformation theory.
This document discusses soil-structure interaction calculations for rigid hydraulic structures using the finite element method (FEM). It examines two common computational models: the Winkler and Pasternak models. The Winkler model represents soil behavior with independent vertical springs, while the Pasternak model adds a shear layer between the soil and structure. Equations are provided for deriving the stiffness matrix of a beam foundation considering each model. The influence of these models on displacements and developed stresses of rigid structures is evaluated through FEM calculations.
As previsões para o futuro incluem: 1) computadores substituirão muitos trabalhos, 2) algas serão cultivadas em larga escala para alimentação e combustível, 3) doenças resistentes a antibióticos se tornarão uma ameaça maior à saúde.
El documento presenta un extracto de la obra "Cartas de amor a Stalin" de Juan Mayorga. En él, Bulgákov lee en voz alta dos cartas que ha escrito dirigidas a Stalin, en las que se queja de que sus obras han sido prohibidas y censuradas en la Unión Soviética. Su esposa intenta consolarlo y hacerle ver que aunque las autoridades lo rechacen, ellos pueden ser felices juntos en su casa y en Rusia.
Design and analysis of laminated composite plates using MATLABkartik kulkarni
This document discusses the analysis of laminated composite plates using finite element analysis in ANSYS and classical lamination theory calculations in MATLAB. It analyzes simply supported symmetrically laminated composite plates subjected to uniform loads. First, a 4-ply composite plate is modeled and analyzed in ANSYS, and the results are validated with MATLAB calculations. Different fiber orientations are then investigated. Next, a 20-ply plate is modeled and analyzed, revealing that bending stiffness values D16 and D26 become insignificant for thicknesses over 16 plies. The document concludes that fiber orientation, stacking sequence, and number of plies all affect deflection results.
This document presents the development of a state space equation approach to obtain the three-dimensional solution for thick orthotropic plates with symmetric clamped-free edges. The state space method allows for an exact solution that satisfies all equations of elasticity and accounts for all elastic constants. The system matrix, which is a key part of the state space solution, is derived for symmetric clamped-free boundary conditions. Modal expansions are used to develop sixth-order differential equations governing the transverse displacement, whose solutions provide the stresses and displacements across the plate thickness.
Kantorovich-Vlasov Method for Simply Supported Rectangular Plates under Unifo...IJCMESJOURNAL
In this study, the Kantorovich-Vlasov method has been applied to the flexural analysis of simply supported Kirchhoff plates under transverse uniformly distributed load on the entire plate domain. Vlasov method was used to construct the coordinate functions in the x direction and the Kantorovich method was used to consider the assumed displacement field over the plate. The total potential energy functional and the corresponding Euler-Lagrange equations were obtained. This was solved subject to the boundary conditions to obtain the displacement field over the plate. Bending moments were then obtained using the moment curvature equations. The solutions obtained were rapidly convergent series for deflection, and bending moments. Maximum deflection and maximum bending moments occurred at the center and were also obtained as rapidly convergent series. The series were computed for varying plate aspect ratios. The results were identical with Levy-Nadai solutions for the same problem.
Three dimensional static analysis of two dimensional functionally graded platesijmech
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated. The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it shows very good agreement.
Three dimensional static analysis of two dimensional functionally graded platesrtme
In this paper, static analysis of two dimensional functionally graded plates based on three dimensional
theory of elasticity is investigated. Graded finite element method has been used to solve the problem. The
effects of power law exponents on static behavior of a fully clamped 2D-FGM plate have been investigated.
The model has been compared with result of a 1D-FGM plate for different boundary conditions, and it
shows very good agreement
This summary analyzes the free vibration of laminated composite and sandwich plates using the Euler-Lagrange equation based on first order shear deformation theory. The document presents analytical formulations and solutions for the natural frequency of simply supported composite and sandwich plates. The results are compared to previous literature. The theoretical model accounts for transverse shear deformation, transverse normal strain/stress, and nonlinear variation of in-plane displacement through the thickness, modeling warping more accurately without shear correction coefficients.
A COMPARATIVE STUDY OF VARIOUS METHODS TO EVALUATE IMPEDANCE FUNCTION FOR SHA...Samirsinh Parmar
Impedance function, Foundation Vibration, dynamic soil-structure interaction, Barkan, Dominguez, Dobry and Gazetas for evaluation of impedance functions for various modes of vibration of shallow foundation
This document presents a higher-order finite element model to predict the linear buckling behavior of laminated beam structures. The model is based on a displacement field that assumes a non-linear variation of displacements through the beam width and depth, eliminating the need for shear correction factors. Numerical applications show the model accurately predicts buckling loads for isotropic and anisotropic beams of various cross-sections when compared to other models and closed-form solutions. The higher-order model is more versatile and accurate than lower-order models like Euler-Bernoulli for a range of slenderness ratios and beam geometries.
This document analyzes the deflections and stresses in composite laminated plates using the dynamic relaxation method. It summarizes the following key points:
1) The dynamic relaxation method is used to solve the equilibrium equations for laminated plate deformation. This method converts the equations to dynamic form by adding damping and inertia terms.
2) Factors like shear deformation, length-to-thickness ratio, number of layers, material anisotropy, and fiber orientation affect plate deflection based on the analysis of square laminated plates with uniform loading.
3) Linear analysis overpredicts deflection compared to the dynamic relaxation method, which accounts for large deflections. Deflection depends on load size, ply orientation, and other
IRJET- Comparison Between Thin Plate and Thick Plate from Navier Solution us...IRJET Journal
This document compares thin plate theory and thick plate theory using MATLAB software. It derives expressions for bending moments, twisting moments, and shear forces for both Kirchhoff's thin plate theory and Reissner's thick plate theory. Detailed programs are written in MATLAB to analyze rectangular plates of varying thickness under Navier's solution. The results show that stress resultants from thin plate theory are independent of thickness, while thick plate theory accounts for thickness in its expressions. Plots demonstrate how bending moments, twisting moments, and shear forces vary across plates of different thicknesses according to each theory.
The document discusses analytical expressions for notch stress intensity factors (NSIFs) under torsion loading for circumferentially-notched rounded bars. It begins by reviewing previous work on NSIFs and stress fields near notches and cracks. It then presents an exact, closed-form solution for the mode III NSIF of a deep V-notch and extends this solution to finite notches using a shape function. The goal is to determine NSIFs directly from theoretical stress concentration factors in order to better understand notch mechanics.
Reliability Analysis of the Sectional Beams Due To Distribution of Shearing S...researchinventy
This paper shows the results of the Reliability Analysis of the sectional beams due to distribution of Shear Stress. It is assumed that the load was uniformly distributed over the beam. It is discussed that the distribution of shear stress over the beam. It is discussed that the average shears stress and maximum shear stress across the section of the beam for Weibull distribution. The reliability analysis of distribution of shearing stresses over sectional beams is performed. Also it is derived that the hazard functions for these types of beams. Reliability comparison has also been done for the sectional beams. It is observed that the reliability of the beam decreased when the width (b) of the beam decreases, and the load (F) is high. The reliability of the beam is increased when the height (h) of the triangular section increases , diameter(d) of the circular beam is increased and parameter 푘 decreasses
Analysis of stiffened plate using FE ApproachIJMER
The objective of the present investigation is to study the strengthening effect of the stiffeners
on the buckling of unperforated and perforated plate when they are reinforced in longitudinal and
transverse directions. The plate is subjected to inplane uniform uniaxial end compression load having
simply supported plate boundary condition. The parameters considered are plate aspect ratio, area ratio and types of stiffeners. The analysis has been carried out using ANSYS finite element software. The buckling analysis shows that the influence of transverse stiffener is less when compared to longitudinal stiffener
An axisymmetric bending and shear stress analysis of of functionally graded ...eSAT Journals
Abstract The effects of exposure from electromagnetic radiations of wireless cellular transmission towers to human health have attracted the attention of many researchers. Different works have revealed the harmful of electromagnetic radiation exposure to human health based on distance from the source and period of exposure. As one stays closer and at a pro-longed period from the transmission sites, the possibility of being affected by the radiation source becomes higher. In this work, we review some of the works on assessment of electromagnetic radiation exposure and propose measures for determining safety zones based on the cases of cellular transmission towers in the Tanzania environment to avoid extended exposure to electromagnetic radiation. Key words- Cellular transmission towers; Electromagnetic radiations; Health effects; Exposure limits
APPLICATION OF THE BOUNDARY ELEMENT METHOD IN DETERMINING THE CRITICAL BUCKLI...Engenheiro Civil
The boundary element method (BEM) is used in this work to obtain the critical buckling parameters of
perforated plates. The plates have square geometry with central square hole. Compression is unidirectional and
uniformly applied at opposite edges. The values for buckling parameters are obtained for many values of thickness.
The shear deformation effect is included in the bending of plates using the isotropic model. The effect of geometric
nonlinearity is introduced by adding two integral in the formulation of the BEM: one is applied in the domain and the
other in the contour. The boundary integral can be related to one of the natural conditions according to the boundary
value problem. Quadratic continuous and discontinuous boundary elements were used. The source points were
positioned on the boundary. The singularity subtraction technique and the transformation of variables were used for
the Cauchy and weak type of singularities, respectively, when the integration is performed in elements containing the
source point. Rectangular cells were used to discretize the domain integral related to the geometric nonlinearity effect.
The results were compared with other authors.
This document provides an overview of plate bending theory. It discusses how plate theory models bending in thin plates using Kirchhoff's plate theory and thick plates using Mindlin plate theory. Kirchhoff's plate theory assumes plates bend without shear deformation and strains are related to plate deflection. Mindlin plate theory allows for shear deformation and separates plate rotation and deflection. The document also discusses deriving plate stresses and forces, applying boundary conditions, and using triangular plate bending elements in finite element analysis.
This document summarizes key concepts in plate theory from Chapter 13. Some key points:
- Flat plates are structural members where the thickness is small relative to length and width, with the middle surface lying in a plane. Plate theory reduces the 3D problem to 2D.
- Stress resultants in plates include in-plane forces (Nxx, Nyy, Nxy), bending moments (Mxx, Myy), and shears (Qx, Qy). Boundary conditions for plates include no displacement, no slope, no moment, and no shear.
- Closed-form solutions exist for rectangular and circular plates with various edge conditions (simply supported, fixed) subjected to uniform loads
International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document presents a third order shear deformation theory to analyze flexure of thick cantilever beams. The theory uses a sinusoidal function in the displacement field to account for transverse shear deformation effects through the beam thickness. Governing equations and boundary conditions are derived using the principle of virtual work. Numerical examples of a cantilever beam with a cosine load distribution are presented and displacement, stress results are obtained in non-dimensional form. The results are discussed and compared to other beam theories to demonstrate the efficiency of the third order shear deformation theory.
This document discusses soil-structure interaction calculations for rigid hydraulic structures using the finite element method (FEM). It examines two common computational models: the Winkler and Pasternak models. The Winkler model represents soil behavior with independent vertical springs, while the Pasternak model adds a shear layer between the soil and structure. Equations are provided for deriving the stiffness matrix of a beam foundation considering each model. The influence of these models on displacements and developed stresses of rigid structures is evaluated through FEM calculations.
As previsões para o futuro incluem: 1) computadores substituirão muitos trabalhos, 2) algas serão cultivadas em larga escala para alimentação e combustível, 3) doenças resistentes a antibióticos se tornarão uma ameaça maior à saúde.
El documento presenta un extracto de la obra "Cartas de amor a Stalin" de Juan Mayorga. En él, Bulgákov lee en voz alta dos cartas que ha escrito dirigidas a Stalin, en las que se queja de que sus obras han sido prohibidas y censuradas en la Unión Soviética. Su esposa intenta consolarlo y hacerle ver que aunque las autoridades lo rechacen, ellos pueden ser felices juntos en su casa y en Rusia.
El documento presenta una breve introducción a los componentes electrónicos, incluyendo resistencias, que se oponen al paso de corriente eléctrica de acuerdo a la ley de Ohm, condensadores que almacenan energía eléctrica, y alternadores y dinamos que generan corriente eléctrica.
El documento describe los aspectos clave de un supermercado típico. Ofrece bienes de consumo a bajo precio mediante el sistema de autoservicio, donde los clientes eligen libremente los productos mientras navegan por las secciones organizadas del supermercado. Los supermercados buscan maximizar las ventas aplicando conocimientos de psicología del consumidor para influir en la navegación y compra de los clientes.
Novedades de la versión 7.5 de TopSolid'Camtecnocim
El documento resume las principales novedades de TopSolid'Cam 7.5, incluyendo la capacidad de editar áreas de mecanizado de forma asociativa, planeado manual basado en esbozos, super acabado 3D, rotura de esquinas con herramientas de bola, documentación para taller, uso de variables en cuadros de diálogo, operaciones sin herramienta asignada, herramientas de torneado flash y utilización de los postprocesadores de la versión 6.
Los estudiantes escribieron historias breves ampliando y ordenando frases sobre sus casas, sillas, pizarras, instrumentos musicales y mesas. Las historias varían en detalles pero generalmente describen una casa grande, una silla pequeña en la clase, escribir en la pizarra, el órgano como instrumento favorito y una mesa cuadrada.
Este documento presenta el boletín electrónico "Rockstalgias", el cual contiene varios artículos y secciones sobre literatura y música rock. El artículo principal analiza los orígenes de la literatura de rock en Cuba, mencionando algunos escritores pioneros como Abel Prieto, Reynaldo Montero y Sergio Cevedo Sosa que incluyeron referencias al rock en sus obras a partir de los años 1960 y 1970 a pesar de la censura. Otras secciones incluyen fragmentos de novelas, cuentos, poemas, efemérides
Valoración económica del embalse de Biscarrués elaborada por el profesor de Análisis Económico de la Facultad de Económicas de la Universidad de Zaragoza y socio fundador de la FNCA, Pedro Arrojo.
La enfermedad de Newcastle aviar es causada por un paramixovirus que causa grandes pérdidas económicas en la industria avícola. El virus puede presentarse en diferentes cepas con distintos niveles de virulencia, desde cepas lentogénicas hasta cepas altamente patógenas que causan una enfermedad aguda y letal. La clasificación, signos clínicos, vacunación e inmunidad contra la enfermedad de Newcastle son importantes para controlar esta enfermedad viral altamente contagiosa que afecta a much
A closed form solution for stress concentration around a circular hole in a lIAEME Publication
1) The document presents a closed-form solution to determine stress concentration around a circular hole in an infinite plate with linearly varying stress.
2) The equation developed can determine stress field around the hole without need for computationally intensive numerical methods.
3) Results from the closed-form solution are compared to finite element analysis and show close agreement.
This document describes a finite element analysis of a thin plate with different hole geometries under tension. Circular, elliptical, and rectangular holes were modeled in the center of a plate. Theoretical stress concentrations were calculated and compared to FE model results. A convergence study determined an optimal mesh size of 25mm. Results from full and quarter plate models showed good agreement, with stress value deviations within 2% and displacement deviations within 1%.
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This chapter discusses stress and strain in materials subjected to tension or compression. It defines stress as the load applied over the cross-sectional area. Strain is defined as the change in length over the original length. Hooke's law states that stress is proportional to strain for elastic materials. Young's modulus is the constant of proportionality between stress and strain. The chapter also discusses stress and strain calculations for materials with non-uniform cross-sections, as well as examples of stress and strain problems.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
This document summarizes an article that presents an analytical model for the electromechanical dynamics of a simply-supported micro-plate subjected to electrostatic excitation. The model derives equations of motion for the plate and uses Galerkin's method to obtain closed-form solutions for static deflection and natural frequencies. The effect of applied voltage, gap height, and plate dimensions on natural frequencies is illustrated. A reduced-order model is also developed and compared to the distributed parameter model.
This document presents the analysis of non-lifting potential flow past a thin symmetric hydrofoil using a finite difference method. The objectives are to solve the potential flow problem around a 2D hydrofoil and calculate the pressure distribution. A NACA 0012 hydrofoil is used. Governing equations and boundary conditions for the potential flow are described. A structured algebraic grid is used to generate points around the hydrofoil. Finite difference equations are derived and discretized on the grid points. The pressure coefficient is then calculated and compared to experimental results, showing good agreement.
An Analytical Method for Static Earth Pressure Distribution against Rectangul...IJERA Editor
Analytical methods for computing the lateral earth pressure against tunnel is vastly used by engineers all over the
world. Conventional analytical methods compute the lateral pressure in either active or passive state while the
stress state usually falls between these two boundaries in many practical cases. Furthermore, using these
boundary coefficients lead to either overestimated or underestimated results in design. Thus, a modified method
based on the strain increment theory for calculating the lateral pressure against rectangular tunnels is presented
herein to consider the amount of lateral deformation at each depth. First, the results for different values of
overburden depth, friction angle and wall mobilized angle are investigated. Then comparative finite element
analyses were performed to examine the effectiveness of the method. According to this study, the pressure
pattern is completely nonlinear especially at the corners of tunnel lining. In fact, the pressure increases
nonlinearly to about three times of the value at top. Lateral earth pressure decreases with the increase of friction
angle which is in good agreement with finite element results. Overall, the pressure patterns derived by this
method for shallow depths (less than tunnel height) are almost the same as those computed by finite element
method.
This document provides an overview of geomechanics concepts for petroleum engineers. It discusses stress and strain theory, elasticity, homogeneous and heterogeneous stress fields, principal stresses, and the Mohr circle construction. It also covers rock deformation mechanisms including cataclasis and intracrystalline plasticity. Key concepts are defined such as normal and shear stress, elastic moduli like Young's modulus and Poisson's ratio, elastic stress-strain equations, and strain measures including conventional, quadratic, and natural strain.
This document describes a study that uses finite element modeling to analyze heat transfer and thermal stresses in a circular plate due to radiation. The study solves the heat conduction equation using finite element methods to determine transient temperature distributions in the plate. Boundary conditions include radiation on the upper surface, a constant temperature on the lower surface, and insulation on the circular edge. Results for temperature, displacement, and thermal stresses in the plate are determined numerically and graphically. The finite element model provides more accurate solutions than previous analytical methods for problems with complex geometries and boundary conditions.
This document presents a study analyzing the static flexural behavior of thick isotropic beams using a higher order shear deformation theory. The theory considers axial displacement as consisting of elementary bending theory displacement plus a hyperbolic shear deformation term. Governing differential equations are derived through the principle of virtual work. Numerical results for axial displacement, transverse displacement, stresses are obtained for cantilever beams under transverse loading and compared to other theories.
This document describes a study that uses COMSOL Multiphysics software to model squeeze-film gas damping in a piezoelectric (PZT) accelerometer. The model represents the solid parts of the accelerometer and solves for the gas pressure distribution using the Reynolds equation. Simulation results show the deformation of the proof mass at different ambient pressures, with higher pressure resulting in greater damping and smaller oscillations. Analysis of the displacement over time indicates squeeze-film gas damping is an important consideration in MEMS device design.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
The document presents an analytical method for determining stress distributions in pin-loaded orthotropic plates. The method assumes a rigid pin, clearance between the pin and plate, and a constant coefficient of friction. Numerical results are shown for normal, tangential and shear stresses on the cavity for different composite layups. The method can predict stresses for varying clearances and a perfectly fitting pin. Experimental validation and improvements to contact modeling are recommended.
1. The document introduces the concepts of stress and strain in rheology, the science of deformation and flow of materials. It discusses how stress is quantified by factors like force and pressure.
2. The lecture defines stress as a dynamic quantity expressing force magnitude, and strain as a kinematic quantity expressing media deformation. It explains how to describe an object's complete stress state in 3D and determine if stress will cause failure.
3. Key concepts covered include surface forces vs. body forces, tractions as normalized forces, and using the Cauchy stress tensor to represent the full stress state at a point and find stress on any plane.
Final Report Turbulant Flat Plate AnsysSultan Islam
- The document describes a computational fluid dynamics (CFD) simulation of turbulent flow over a flat plate using ANSYS CFX.
- The simulation aims to validate results against experimental data from NASA and analyze sensitivity of skin friction coefficient and velocity profiles.
- The flat plate geometry, meshing approach, and boundary conditions are described based on the NASA and Caelus experiments.
- Results for velocity profiles and skin friction coefficients along the plate are presented and validated against experimental trends.
- Grid convergence and sensitivity to turbulence models are analyzed, with the SST and k-epsilon models showing similar results.
MODELING OF MANUFACTURING OF A FIELDEFFECT TRANSISTOR TO DETERMINE CONDITIONS...antjjournal
In this paper we introduce an approach to model technological process of manufacture of a field-effect
heterotransistor. The modeling gives us possibility to optimize the technological process to decrease length
of channel by using mechanical stress. As accompanying results of the decreasing one can find decreasing
of thickness of the heterotransistors and increasing of their density, which were comprised in integrated
circuits.
On Analytical Approach to Prognosis of Manufacturing of Voltage Divider Biasi...BRNSS Publication Hub
In this paper, we introduce an approach for prognosis of manufacturing of voltage divider biasing common emitter amplifier based on bipolar transistors with account mismatch-induced stress. Based on this prognosis, we formulate some recommendations for optimization of manufacturing of the amplifier. Main aims of the optimization are as follows: (1) Decreasing dimensions of elements of the considered operational amplifier and (2) increasing of performance and reliability of the considered bipolar transistors. Dimensions of considered bipolar transistors will be decreased due to manufacture of these transistors framework heterostructure with specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects. Performance and reliability of the above bipolar transistors could be increased by optimization of annealing of dopant and/or radiation defects and using inhomogeneity of the properties of heterostructure. Choosing of inhomogeneity properties of heterostructure leads to increasing of compactness of distribution of concentration of dopant. At the same time, one can obtain increasing of homogeneity of the above concentration. In this paper, we also introduce an analytical approach for prognosis of technological process of manufacturing of the considered operational amplifier. The approach gives a possibility to take into account variation of parameters of processes in space and at the same time in space. At the same time, one can take into account nonlinearity of the considered processes.
This document analyzes the effect of different directrix shapes (circular, parabolic, elliptical, inverted catenary) on membrane stresses in cylindrical shell roofs. Membrane theory is used to determine the normal forces (Nx, Nθ, Nxθ) and stresses under self-weight and live loads. Equations for each directrix are presented. An example problem is solved and results are shown in tables comparing stresses for each directrix. The analysis found that stresses are lowest with an inverted catenary directrix and highest with an elliptical directrix to cover the same area.
Crack problems concerning boundaries of convex lens like formsijtsrd
The singular stress problem of aperipheral edge crack around a cavity of spherical portion in an infinite elastic medium whenthe crack is subjected to a known pressure is investigated. The problem is solved byusing integral transforms and is reduced to the solution of a singularintegral equation of the first kind. The solution of this equation is obtainednumerically by the method due to Erdogan, Gupta , and Cook, and thestress intensity factors are displayed graphically.Also investigated in this paper is the penny-shaped crack situated symmetrically on the central plane of a convex lens shaped elastic material. Doo-Sung Lee"Crack problems concerning boundaries of convex lens like forms" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11106.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/11106/crack-problems-concerning-boundaries-of-convex-lens-like-forms/doo-sung-lee
Driving Business Innovation: Latest Generative AI Advancements & Success StorySafe Software
Are you ready to revolutionize how you handle data? Join us for a webinar where we’ll bring you up to speed with the latest advancements in Generative AI technology and discover how leveraging FME with tools from giants like Google Gemini, Amazon, and Microsoft OpenAI can supercharge your workflow efficiency.
During the hour, we’ll take you through:
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This webinar is ideal for professionals seeking to harness the power of AI within their data management systems while ensuring high levels of customization and security. Whether you're a novice or an expert, gain actionable insights and strategies to elevate your data processes. Join us to see how FME and AI can revolutionize how you work with data!
GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
Leonard Jayamohan, Partner & Generative AI Lead, Deloitte
This keynote will reveal how Deloitte leverages Neo4j’s graph power for groundbreaking digital twin solutions, achieving a staggering 100x performance boost. Discover the essential role knowledge graphs play in successful generative AI implementations. Plus, get an exclusive look at an innovative Neo4j + Generative AI solution Deloitte is developing in-house.
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
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DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
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GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
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AI 101: An Introduction to the Basics and Impact of Artificial Intelligence
Fa3110171022
1. by, J. Carrera, J.E. Martínez and L. Ferrer / International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1017-1022
3D Stress Distribution Analysis around Blind-holes in Thin
Plates Under Non-uniform Tension Loads
by, J. Carrera, J.E. Martínez and L. Ferrer
PhD, Professor of Mechanical and Industrial Engineering Division, Engineering Faculty, Universidad
NacionalAutónoma de México, Distrito Federal, México.
PhD, PostgraduateDivision, EngineeringFaculty, Universidad Nacional Autónoma de México, Distrito Federal,
México.
PhD, late Professor, Mechanical and Industrial EngineeringDivision, EngineeringFaculty, Universidad Nacional
Autónoma de México, Distrito Federal, México
ABSTRACT.
A three-dimensional photo elasticity created to fasten different parts of the component, or
analysis and an interactive finite-element package simply by design. The determination of the stress
were used in parallel to analyze the stress concentration around holes and notches was, and
distribution on the root of blind-holes in thin still is, one of the most common applications of 2D
plates. Experimental analysis was conducted via and 3D photoelasticity of machine elements design.
stress freezing method. The FEM analysis was In the literature, theoretical, experimental and
performed with the ABAQUS commercial code numerical solutions were obtained for several cases
using material properties obtained where a circular hole has a particular function in thin
experimentally as input. The results showed that or thick plates.Iancuet. al. [2] evaluated the stress
the maximum stress distribution occurred in distribution inside thick bolted plates along the
three zones: the first one at the beginning of the bearing plane normal to theplate surface.
blind-hole; the second one at the transition zone Their experimental and numerical results help them
where the root begins his shape; and at the center to develop an improved joint design.
of the root.These results are expected to improve Another stress distribution research in plates with
blind-hole design according to its function. The circular holes was developed by Wang et. al. [3]. In
combined use of experimental and numerical their paper, they proposed a replaced superposition
methods provides more information than each method to reconstruct experimental photoelastic
method taken alone. This information is essential fringe patterns of a near-surface circular hole.
when the relation between depth and thickness Peindlet. al. [4] did an analysis of a total shoulder
has to be taken into account. As shown here, the replacement system via photoelastic stress freezing.
stresses near the free boundary are relevant for They showed that maximum stress occurred at the
failure considerations, for example due to the neck and at the component-bone interface beneath
presence of debris in thin plates or sheets. An simulated PMMA inclusions on both axial and
analog statement can be made for blind-holes coronal planes. Those planes exhibit blind-holes for
made to fasten metal sheets with bolts or hollows different depths and root-shapes due to the whole
for human prosthesis. replacement system.
KEY WORDS- Blind-hole, stress-freezing method, However, to the best of our knowledge, no
FEM, photoelasticity, stress concentration. experimental-numerical solution of the stress
distribution around a blind-hole when a thin plate is
I. Introduction under tension loads available in the literature. Figure
In engineering practice the presence of 1 illustrates this point.
holes, notches or any other geometry that causes The purpose of this paper is three-fold. First to
stress concentration has been studied since Kirsch reproduce the boundary conditions required at the
[1] in 1898. Problems with abrupt change of plate and develop a model whereby stress freezing
geometry in structural components are well known: method is used. Second, to analyze and acquire new
stress concentration may be induced around the hole information about maximal stresses along blind-
or notch, causing severe reductions of the strength hole's route and root. And third, to show how
and fatigue life of a structure. An important technical numerical results corroborate and support qualitative
example of this situation is given when debris are results from the experimental ones.
present in laminating processes; such defects can be
well modeled as blind holes. Many are the cases
when the structural component has blind-holes
1017 | P a g e
2. by, J. Carrera, J.E. Martínez and L. Ferrer / International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1017-1022
II. Mathematical statement of the
problem 𝑎𝑡 𝑦 = 𝐴 ∶ 𝐮 𝕡, 𝑡 = 0 3
Consider a homogeneous, isotropic, elastic body
Ω ⊂ ℝ3 at equilibrium define by |x| < L, |y| < A , |z| 𝑎𝑡 𝑧 = 0,2 ∶ 𝐮 𝕡, 𝑡 = 0 4
<2hwhere 𝐿, 𝐴, > 0 are given. It contains a
cylindrical blind-hole of radius a whose generators for every 𝕡 ∈ 𝜕Ω,n being a unit normal vector at 𝕡,
are perpendicular to the bounding planes and and 𝑓 𝑦 is a singularity function [11] called the unit
consists of two geometries, a cylinder ramp starting at 𝑦 𝐴 = 𝜎.
2
𝐶𝑙 = 𝑥, 𝑦, 𝑧 𝑥 2 + 𝑦 2 ≤ 𝑎2 , 0 ≤ 𝑧 ≤ 𝑐1 III. Experimental Approach
The employed loading system could
and a semi-sphere SE produce only compression forces; we need to carry
out a new system in order to generate the wanted
𝑆𝐸 loadon 𝜕Ω. Figure 2 illustrates the procedure that the
= 𝑥, 𝑦, 𝑧 𝑥 2 + 𝑦 2 + 𝑧 − 𝑐1 2
≤ 𝑟 2 ; 𝑐1 ≤ 𝑧 ≤ 𝑐1 + 𝑟 authors propose to produce the same system as
shown in Fig. 1. If a plate is subjected to a central
so the body Ω = 𝑅𝐸𝐶 ∖ 𝐶𝑙 ∪ 𝑆𝐸 , where 𝑅𝐸𝐶 is the load, as illustrated in Fig. 2a, the bending stresses
rectangle of dimensions 𝐿 × 𝐴 × 𝑡. are of equal magnitude at the top and bottom of the
For this particular study the plate has the dimensions: beam (compression at the top and tension at the
𝐿 = 60 mm 2.36 in , 𝐴 = 26 mm 1 in , 𝑎 = bottom). The stress distribution over the cross
2.4 mm 0.1 in , 𝑟 = 1.6 mm 0.06 in , 𝑡 = section is shown in Fig. 2a (right). In order to have a
9.2 mm 0.36 in . stress from a constant value 𝜎 (top) to a constant
Let the plate be subjected to a non-uniform tensile value 𝜎 𝑚𝑎𝑥 (bottom) at the plate, a rectangular
load along the x-axis and parallel to the bounding groove was made long enough to have a shorter plate
planes (Figure 1). below the groove, as shown in Fig. 2b.
Figure 1 Geometrical configuration of a plate weakened by
a circular blind-hole
Figure 2 Stress distribution diagrams in the plate with
The linear system of partial differential equations for the groove and acting load.
the fieldu, E and S is
The shear force has the constant magnitude 𝐹 2
1 between the load and each support. The shear stress
𝐄 𝕡, 𝑡 = 1 + 𝜈 𝐒 𝕡, 𝑡 − 𝜈 tr𝐒 𝕡, 𝑡 𝐈 is maximum at the mid-height of the plate and
𝐸
1 1 decreases parabolically as it reaches the bottom of
𝐄 𝕡, 𝑡 = 𝛁𝐮 𝕡, 𝑡 + 𝛁𝐮 𝕡, 𝑡 T the plate; in coordinates,
2
Div𝐒 𝕡, 𝑡 + 𝐛 𝐨 𝕡, 𝑡 = 𝜌 𝑜 𝐮 𝕡, 𝑡 2
𝐹 2 𝐻
𝜏 𝑥𝑦 = − 𝑦1 2 5
whereE is called the infinitesimal strain field, S is the 𝑚𝑎𝑥 2 𝑡𝐻 3 12 2
first Piola-Kirchhoff stress field, 𝐛 𝐨 is the body force,
u is the displacement field and 𝐸, 𝜈 are called The bending and shear stresses are of comparable
Young’s modulus and Poisson’s ratio respectively magnitude if and only if W and H are of the same
[5]. magnitude. Since for this particular problem 𝑊 =
280 𝑚𝑚 (10.9 𝑖𝑛), 𝐻 = 62 𝑚𝑚(2.4𝑖𝑛) from [11]
As to the boundary conditions, it is required that
𝜏 𝑥𝑦 1 𝐻
𝑚𝑎𝑥
𝜕𝐮 𝕡, 𝑡 = 6
𝑎𝑡 𝑥 = 𝐿∶ = 𝑓 𝑦 2 𝜎𝑥 𝑚𝑎𝑥 2 𝑊
𝜕𝐧
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3. by, J. Carrera, J.E. Martínez and L. Ferrer / International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1017-1022
the ratio between the maximum bending and shear Two fasteners were made of aluminum alloy,
stress in the plate will be 17.5:1 respectively. 2070-T6, with mechanical properties:𝐸 =
Having the groove offers one particular 72 GPa 10.34 Mpsi , 𝜈 = 0.32to support the plate
advantage: block the way of every fringe pattern during the loading process. Loading was
produced by the loading cell so we can isolate the implemented by a load cell system instrumented
part under tension that is the analyze one. with strain gages and a meter to read the force
Once the rectangular groove was made, a blind- transmitted to plate.
hole at the mid-height and mid-width of the plate was
drilled. Finally there is a plate with a centered The specimens were loaded at room temperature
circular blind-hole subjected to a non-uniform tensile and subjected to a conventional stress-freezing cycle
load as shown in Fig. 2c and Fig. 1. with a critical temperature of115°𝐶 240°𝐹 .
IV. Experimental Method
Stress freezing method was used to determine
the stress distribution in the transversal and
longitudinal planes at blind-hole´s neighborhood.
Once the slices were obtained, a circular polariscope
was employed to analyze the fringe patterns.
As shown in the Fig. 3, transversal and
longitudinal slices were cut out from the plate to be
analyzed separately. Circular polariscope was used
because the circular polarization eliminates isoclinics
(loci of points of constant inclination of the principal
axes of refraction).
Model Fabrication
A508 × 508 × 9.6 mm 20 × 20 ×
0.38 incommercial photoelastic sheet, PSM-9, was
supplied by Photoelastic Division of Measurements Figure 3 Slicing at the blind-hole´s neighborhood
Group, Inc. The room- temperature stress-optic along planes of principal stress
coefficient, C, for PSM-9 is nominally10.5 kPa/
𝑓𝑟𝑖𝑛𝑔𝑒/𝑚. A950 N 212.5 lb force was produced by the cell
At the stress-freezing temperature load system to the plate once the critical temperature
of110°𝐶 𝑡𝑜 120°𝐶 230°𝐹 𝑡𝑜 250°𝐹 , the stress- had been reach. The whole process was performed
optic coefficient is approximately 0.50 kPa/ exactly as Dally and Riley [7] appoint.
𝑓𝑟𝑖𝑛𝑔𝑒/𝑚 [10]. After stress freezing, the model was rough cut on
Eighteen280 × 62 × 9.6 mm 11.02 × 2.44 × a band saw and ground to a final thickness of:
0.38 inpieces were cut out from the commercial i)3.1 mm 0.23 in for cross-sectional slice, and
sheet. A blind-hole of4.8 mm 3 16 in in diameter ii)2.9 mm 0.11 in for longitudinal slice. Air jet-
with cooling was used to prevent local heating.
three3.175, 6.41and7.9 mm 1 8, 1 4, 5 16 in diff
erent depths was drilled at 140 mm 5.5 in from the
right side and13 mm 0.51 in from the bottom of the Finite Element Analysis
plate. A90 × 7.9 × 9.6 mm 3.5 × 0.31 ×
This study was performed using the finite
0.38 inrectangular groove was drilled at40 mm 1.57 element code ABAQUS 6.7-1 [12]. The finite
infrom the left side and31 mm1.22 infrom the
element model was built with the same finite
bottom of the plate, as in Figure 2b.
element code and the postprocessor used to view the
results was ABAQUS/Viewer 6.7-1. A typical mesh
All the pieces were cut out from the commercial of a plate with a circular blind-hole is shown in Fig.
sheet with a band-saw operated at medium speed. 4.
The groove was made with an end mill with two To apply the boundary conditions, three areas
fluted solid-carbide milling cutters with spiral flutes. were created to suffice this requirement. A first one
Blind hole was drilled with a carbide-tipped boring of 1.5 × 9.6 mm 0.06 × 0.38 in to apply constant
tool at a medium speed (1280 rpm). pressure. Then the second and third ones of1 ×
9.6 mm 0.04 × 0.38 in were created to apply
Loading Conditions displacement restrictions about y and z axis.
The plate is divided into two parts: a rectangle
corresponding to the one containing the blind-hole,
1019 | P a g e
4. by, J. Carrera, J.E. Martínez and L. Ferrer / International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1017-1022
shown in the Fig. 2c with a finer mesh surrounding
the blind-hole and the second part corresponding to whereN is the fringe order, 𝑓𝜎 the material stress-
the rest of the plate with a coarser mesh. 48 elements fringe value and 1 the slice thickness.
were used around the blind-hole. Figure 7a, 7b and 7c shows the FEManalysis
Before the photoelastic stress-freezing test was results. The maximum stress occurs at points 10 and
performed a finite-element solution was obtained for 11, with magnitudes of 25.6 MPa 3.70 ksi and
the case of uniform pressure on the outer boundary 28.3 MPa 4.09 ksi respectively for the first case.
(top rectangle) using the mechanical properties of the For the second and third cases it is more evident
photoelastic material under stress-freezing that the maximum stress occurred all along the blind-
conditions. The main purpose of this analysis was to hole’s route, having his higher value at points
estimate the amount of load to be applied on the 8 31.92 MPa 4.61 ksi and
model and to obtain an estimate of the 11 34.22 MPa 4.95 ksi .
displacements.
Figure 5 Loci of points considered for each slice
From Figure 7 one can conclude that the fringe
pattern indicates that the stress is more uniformly
Figure 4 Typical mesh of a thin plate with a circular
distributed over the thickness when the blind-hole´s
blind-hole presence
depth is 6.4 mm 0.25 in than the other two cases.
From Figure 7c it is interesting to notice that
Experimental and Numerical Results
point 8 has slightly higher stress value than point 11
Figure 5 shows the loci of points considered in
this study for transversal and longitudinal slices. 34.1 MPa 4.93 ksi 𝑣𝑒𝑟𝑠𝑢𝑠 32.81 MPa 4.74 ksi .
Transversal Slices
Longitudinal Slices
Figure 6 shows photoelastic results on the plate
subjected to 981 N 220.2 lb load. On close Photoelastic results for longitudinal slices are
evaluation we can see that the high-stress shown in Figure 6d, 6e and 6f. They illustrate how
concentrations lie on both the transition zone (point the stress concentration lies at point 15 (root) for all
7) and at the free boundary (point 11). depths. Figure 8 shows more evidently this fact: a
The stress plotted in Fig. 7 correspond to the longitudinal slice from a plate with a
maximum principal stress, in this case the tangential 4.6 mm 0.19 in circular blind-hole deepness under
stress, at the blind-hole’s boundary. a 2107 N 473 lb load. The fringe pattern tends to
At the boundary, one of the two principal stresses concentrate at point 15.
vanishes, so, in coordinates
𝑁𝑓𝜎
𝜎 𝜃𝜃 = 7
1
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5. by, J. Carrera, J.E. Martínez and L. Ferrer / International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1017-1022
Figure 6 Photoelastic patterns, shown for representative transversal and longitudinal slices for three plates
subjected to a 981 N (220.2 lb) load each one
Figure 7 Distribution of maximum principal stress around the blind-hole under a 937 N (210 lb) load. First
column corresponds to the transversal and longitudinal slices (3.1 mm depth); second and third columns are
their similar for 6.4 mm and 7.9 mm blind-hole’s depth respectively
Durelli and Riley [16, pp. 212-215] found that results within a margin of uncertainty of
maximum stress for a strip with U-Shaped notches 2.5 MPa 0.36 ksi this is a reasonable result.
under axial load occurs at the center line of the notch
with a deviation angle 𝜙. Rubayi and Taft [9] also
obtained the same locus for the maximum stress for a Discussion and Conclusions
thick bar with U-Shaped notches.
The higher stress was 24.1 MPa 3.48 ksi for the Several experimental and numerical results were
second case (blind-hole´s depth of 6.4 mm (0.25 in)). obtained to analyzed and evaluate the behavior of
The difference obtained by photoelasticity and blind-holes in thin plates. The main interests of this
FEM for stress values are about 15-20%. Taking into study were stress concentration fields along the
consideration that photoelastic methods provide
1021 | P a g e
6. by, J. Carrera, J.E. Martínez and L. Ferrer / International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1017-1022
root’s concavity and along blind-hole’s route under References
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For the transversal slice, experimental and FEM Bedürfnisse der Festigkeitslehre," Z.
results showed that the maximum stress fields VereinesDeutscherIng., 42, 797-807 (1898).
occurred when blind-hole’s depth was 2. Iancu, F., Ding, X., Cloud, G. L., Raju, B. B.,
7.9 mm 0.31 in and were located at points 8 and 11 "Three-dimensional Investigation of Thick
(transition zone and at the free boundary plane Single-lap Bolted Joints," Experimental
respectively). The difference between both stresses is Mechanics, 45 (4), 351-358 (2005).
less than 3%. The fact that the maximal stresses were 3. Wang, W. C., Chen, Y-M., Lin, M-S., Wu, C-P.,
almost of the same magnitude at two points was "Investigation of the Stress Field of a Near-
significant. The finding demonstrates the importance surface Circular Hole," Experimental Mechanics,
of knowing that a possible failure (i.e. crack) may 45 (3), 244-249 (2005).
occur at either point, and not necessarily at point 11 4. Peindl, R.D., Harrow, M. E., Connor, P.M.,
where the experimentalist normally could reach. Banks, D.M., D'Allesandro, D.F., "Photoelastic
For longitudinal slices, experimental results Stress Freezing Analysis of Total Shoulder
provide better qualitative understanding of the stress Replacement Systems," Experimental Mechanics,
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The maximum stress occurs at point 15 (root’s 5. Gurtin, E. M., "An Introduction to Continuum
center) as can be seen in Figure 7. The numerical Mechanics, First ed.," Academic Press, New
results matches the fringe pattern obtained via York (2003).
photoelastic method. 6. Durelli, A. J., Phillips, E. A., Tsao, C. H., "An
The experimental array in order to produce the Introduction to the Theoretical and Experimental
requiring load for this study worked as well as we Analysis of stress and Strain, First ed., "McGraw-
could expect for. For the cases when it was difficult Hill Book Co., New York (1958).
to obtain qualitative results from experimental slices, 7. Dally, J. W., Riley, W. F., "Experimental Stress
the FEM is a very practical way to support them. Analysis, Fourth ed.," College House Enterprises
It has been proven that the deeper the blind-hole LLC, Knoxville, TN (2005).
is, stress at points 8 (for transversal slice) and 15 (for 8. Cernosek, J., "Three-dimensional Photoelasticity
longitudinal slice) must be considered. by Stress Freezing, " Experimental Mechanics,
Figure 8 shows the stress concentration factor 20, 417-426 (1980).
behavior along blind-hole’s boundary. These results 9. Rubayi, N. A., Taft, M. E., "Photoelastic Study of
should give for a scientist and engineer better and Axially Loaded Thick-notched Bars,"
reliable information of what happens inside the hole Experimental Mechanics, 377-383 (October
and be aware of any possible failure. 1982).
After 17 iterations, FEM results converged within 10. Vishay Measurements Group Inc., Photoelastic
the expected results. Division: Instruction Bulletin IB-242, Raleigh,
NC (1996).
11. Crandall, S. H., Dahl, N. C., Lardner, T. J., "An
Introduction to the Mechanics of Solids, Second
ed.," McGraw-Hill International Book Co., New
York (1978).
12. ABAQUS, Inc., "ABAQUS user's manual-version
6.7-1," Pawtucket, RI, USA (2008).
13. Gurtin, E. M., "The Mechanics and
Thermodynamics of Continua, First ed., "
Cambridge Press, London (2010).
Figure 8 Shear stress fringe patterns for a 14. Muskhelishvili, N. I., "Some Basic Problems of
longitudinal slice with 2107 N (474 lb) load the Mathematical Theory of Elasticity, Second
ed.," Noordhoff, Ltd., Groningen, Netherlands
Acknowledgments (1963).
The authors wish to express their appreciation to 15. Frocht, M. M., "Photoelasticity," Vol. II, John
Professor James W. Dally for his invaluable Wiley & Sons, New York (1948).
assistance and comments during this investigation. 16. Durelli, A. J., Riley, W. F., "Introduction to
Also we want to thank Vishay Measurement Group, Photomechanics," Prentice Hall(1965)
Inc. Photoelastic Division for their technical support
during and after the purchase process.
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