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
A Displacement-Potential Scheme to Stress Analysis of a Cracked Stiffened Pan...inventionjournals
This paper deals with an efficient analytical scheme for the analysis of stress and displacement fields of boundary-value problemsof plane elasticity with mixed boundary conditions and material discontinuity. More specifically, the mechanical behavior of a stiffened panel with an edge crack is analyzed under the influence of axial loading, using a new analytical scheme. Earlier mathematical models of elasticity were very deficient in handling the practical stress problems of solid mechanics. Analytical methods of solution have not gained that much popularity in the field of stress analysis, mainly because of the inability of dealing with mixed boundary conditions, irregular boundary shapes, and material discontinuity
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.
The convergence and accuracy of the Dynamic Relaxation (DR) solutions for elastic
large deflection response are established by comparison with various exact and approximate
solutions. New numerical results are generated for uniformly loaded square laminated plates
which serve to quantify the effects of shear deformation, length to thickness ratio, number of
layers, material anisotropy and fiber orientation.
It was found that linear analysis seriously over predicts deflection of plates. The shear
deflection depends greatly on a number of factors such as length to thickness ratio, degree of
anisotropy and number of layers. As the degree of anisotropy increases, the plate becomes
stiffer and when it is greater than a critical value, the deflection becomes virtually
independent on the degree of anisotropy. It was also found that deflection of plates depends
on the angle of orientation of individual plies and the size of load applied.
A comprehensive literature review on different theories of laminated plates have been
reviewed and discussed thoroughly. It has been found that there are two main theories of
laminated plates which are known as linear and nonlinear theories. The two theories are
depending on the magnitude of deformation resulting from loading the given plates. The
difference between the two theories is that the deformations are small in the linear theory,
whereas they are finite or large in the nonlinear theory.
In comparisons between FEM and different numerical methods it has been found that
FEM can be considered of acceptable accuracy, and can also be applied to different
complicated geometries and shapes.
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.
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
A Displacement-Potential Scheme to Stress Analysis of a Cracked Stiffened Pan...inventionjournals
This paper deals with an efficient analytical scheme for the analysis of stress and displacement fields of boundary-value problemsof plane elasticity with mixed boundary conditions and material discontinuity. More specifically, the mechanical behavior of a stiffened panel with an edge crack is analyzed under the influence of axial loading, using a new analytical scheme. Earlier mathematical models of elasticity were very deficient in handling the practical stress problems of solid mechanics. Analytical methods of solution have not gained that much popularity in the field of stress analysis, mainly because of the inability of dealing with mixed boundary conditions, irregular boundary shapes, and material discontinuity
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.
The convergence and accuracy of the Dynamic Relaxation (DR) solutions for elastic
large deflection response are established by comparison with various exact and approximate
solutions. New numerical results are generated for uniformly loaded square laminated plates
which serve to quantify the effects of shear deformation, length to thickness ratio, number of
layers, material anisotropy and fiber orientation.
It was found that linear analysis seriously over predicts deflection of plates. The shear
deflection depends greatly on a number of factors such as length to thickness ratio, degree of
anisotropy and number of layers. As the degree of anisotropy increases, the plate becomes
stiffer and when it is greater than a critical value, the deflection becomes virtually
independent on the degree of anisotropy. It was also found that deflection of plates depends
on the angle of orientation of individual plies and the size of load applied.
A comprehensive literature review on different theories of laminated plates have been
reviewed and discussed thoroughly. It has been found that there are two main theories of
laminated plates which are known as linear and nonlinear theories. The two theories are
depending on the magnitude of deformation resulting from loading the given plates. The
difference between the two theories is that the deformations are small in the linear theory,
whereas they are finite or large in the nonlinear theory.
In comparisons between FEM and different numerical methods it has been found that
FEM can be considered of acceptable accuracy, and can also be applied to different
complicated geometries and shapes.
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.
Dynamic Relaxation (DR) method is presented for the geometrically nonlinear laterally loaded,
rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse
shear deformation. A computer program has been compiled. The convergence and accuracy of the DR
solutions for elastic large deflection response are established by comparison with various exact and
approximate solutions. New numerical results are generated for uniformly loaded square laminated
plates which serve to quantify the effects of shear deformation, length to thickness ratio, number of
layers, material anisotropy and fiber orientation.
It was found that linear analysis seriously over predicts deflection of plates. The shear
deflection depends greatly on a number of factors such as length to thickness ratio, degree of
anisotropy and number of layers. As the degree of anisotropy increases, the plate becomes stiffer and
when it is greater than a critical value, the deflection becomes virtually independent on the degree of
anisotropy. It was also found that deflection of plates depends on the angle of orientation of individual
plies and the size of load applied.
First order orthotropic shear deformation equations for the nonlinear elastic bending response of rectangular plates are introduced. Their solution using a computer program based on finite differences implementation of the Dynamic Relaxation (DR) method is outlined. The convergence and accuracy of the DR solutions for elastic large deflection response of isotropic, orthotropic, and laminated plates are established by comparison with various exact and approximate solutions. The present Dynamic Relaxation method (DR) coupled with finite differences method shows a fairly good agreement with other analytical and numerical methods used in the verification scheme.It was found that: The convergence and accuracy of the DR solution is dependent on several factors including boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR large deflection program using uniform finite differences meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads. All the comparison results for simply supported (SS4) edge conditions showed that deflection is almost dependent on the direction of the applied load or the arrangement of the layers
Dynamic Relaxation (DR) method is presented for the analysis of geometrically linear laterally loaded, rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse shear deformations. A computer program has been compiled. The convergence and accuracy of the DR solutions of isotropic, orthotropic, and laminated plates for elastic small deflection response are established by comparison with different exact and approximate solutions. The present Dynamic Relaxation (DR) method shows a good agreement with other analytical and numerical methods used in the verification scheme. It was found that: The convergence and accuracy of the DR solution were dependent on several factors which include boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR small deflection program using uniform meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads in a fairly good accuracy.
First order orthotropic shear deformation equations for the linear elastic bending response of rectangular
plates are introduced. Their solution using a computer program in FORTRAN language based on finite differences
implementation of the Dynamic Relaxation (DR) method is outlined. The convergence and accuracy of the DR
solutions for elastic linear response of isotropic, orthotropic, and laminated plates are established by comparison with
various exact and approximate solutions. The present Dynamic Relaxation method (DR) coupled with Finite
Differences method (FD) shows a fairly good agreement with other analytical and numerical methods used in the
present study. It was found that the DR linear theory program using uniform meshes can be employed in the analysis
of different thicknesses and length to side ratios for isotropic, orthotropic and laminated fibrous plates under uniform
loads in a fairly good accuracy. These comparisons show that the type of mesh used (i.e. uniform or graded) is
responsible for the considerable variations in the mid – side and corner stress resultants. It is found that the
convergence of the DR solution depends on several factors including boundary conditions, meshes size, fictitious
densities and applied load. It is also found that the DR linear theory can be employed with less accuracy in the
analysis of moderately thick and flat isotropic, orthotropic or laminated plates under uniform loads. It is also found
that the deflection of the plate becomes of an acceptable value when the length to thickness ratio decreases. For
simply supported (SS1) edge conditions, all the comparison results confirmed that deflection depends on the direction
of the applied load and the arrangement of the layers.
Dynamic Relaxation (DR) method is presented for the analysis of geometrically linear laterally loaded, rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse shear deformations. A computer program has been compiled. The convergence and accuracy of the DR solutions of isotropic, orthotropic, and laminated plates for elastic small deflection response are established by comparison with different exact and approximate solutions. The present Dynamic Relaxation (DR) method shows a good agreement with other analytical and numerical methods used in the verification scheme.
It was found that: The convergence and accuracy of the DR solution is dependent on several factors which include boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR small deflection program using uniform meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads in a fairly good accuracy.
Linear And Nonlinear Analytical Modeling of Laminated Composite Beams In Thre...researchinventy
The large current development of aerospace and automotive technologies is based on the use of composite materials which provide significant weight savings compared to their mechanical characteristics. Correct dimensioning of composite structures requires a thorough knowledge of their behavior in small as in large deflection.This work aims to simulate linear and nonlinear behavior of laminates composites under threepoint bending test. The used modelization is based on first-order shear deformation theory (FSDT), classical plate theory (CPT) and Von-Karman’s equations for large deflection. A differential equation of Riccati, describing the variation of the deflection depending on the load, was obtained. Hence, the results deduced show a good correlation with experimental curves
Characteristic orthogonal polynimial application to galerkin indirect variati...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Steady State Thermoelastic Problem in an Infinite Elastic Layer Weakened by a...inventionjournals
The paper is concerned with a thermoelastic behavior in an infinite elastic layer of finite thickness with a crack in it lying in the middle of the layer and parallel to the faces of the layer. The faces of the layer are maintained at constant temperature of different magnitude. The problem is a steady state thermoelastic problem. The layer surfaces are supposed to be acted on by symmetrically applied concentrated forces of magnitude with respect to the centre of the crack. The applied concentrated force may be compressive or tensile in nature. The problem is solved by using integral transform technique .The solution of the problem has been reduced to the solution of a Cauchy type singular integral equation, which requires numerical treatment. The stress-intensity factor and the crack opening displacement are determined and thermal effects on various subjects of physical interest are shown graphically
Acta Mater 57 (2009) 559 Investigation Of The Indentation Size EffectDierk Raabe
Investigation of the indentation size effect through the measurement of the geometrically necessary dislocations beneath small indents of different depths using EBSD tomography
A composite material can be defined as a combination of two or more materials that
gives better properties than those of the individual components used alone. In contrast to
metallic alloys, each material retains its separate chemical, physical, and mechanical
properties. The two constituents are reinforcement and a matrix. The main advantages of
composite materials are their high strength and stiffness combined with low density when
compared to classical materials. Micromechanical approach is found to be more suitable for
the analysis of composite materials because it studies the volume proportions of the
constituents for the desired lamina stiffness and strength.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Applications of layered theory for the analysis of flexible pavementseSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Nonlinear Viscoelastic Analysis of Laminated Composite Plates – A Multi Scale...rtme
Laminated composite plates are widely used in modern structures. Resins of composites are almost made of
polymers which show time dependent and and in some cases stress dependent behaviour. In this paper, a
laminated composite plate is analysed using a multiscale method. At first, material properties of a lamina is
obtained using an analytical micromechanical approach called simplified unit cell method (SUCM) and
then in macromechanical level, Generalized Differential Quadrature Method (GDQM) is used to analyse
laminated composite plate. Schapery's integral is used to model nonlinear viscoelastic behaviour of the
matrix. Prony series is considered to define the compliance of matrix. Micromechanical process includes
obtaining overall properties of the composite by SUCM. Both geometrical and material nonlinearity are
taken into account in order to multiscale analysis of laminated composite plate.
Dynamic Relaxation (DR) method is presented for the geometrically nonlinear laterally loaded,
rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse
shear deformation. A computer program has been compiled. The convergence and accuracy of the DR
solutions for elastic large deflection response are established by comparison with various exact and
approximate solutions. New numerical results are generated for uniformly loaded square laminated
plates which serve to quantify the effects of shear deformation, length to thickness ratio, number of
layers, material anisotropy and fiber orientation.
It was found that linear analysis seriously over predicts deflection of plates. The shear
deflection depends greatly on a number of factors such as length to thickness ratio, degree of
anisotropy and number of layers. As the degree of anisotropy increases, the plate becomes stiffer and
when it is greater than a critical value, the deflection becomes virtually independent on the degree of
anisotropy. It was also found that deflection of plates depends on the angle of orientation of individual
plies and the size of load applied.
First order orthotropic shear deformation equations for the nonlinear elastic bending response of rectangular plates are introduced. Their solution using a computer program based on finite differences implementation of the Dynamic Relaxation (DR) method is outlined. The convergence and accuracy of the DR solutions for elastic large deflection response of isotropic, orthotropic, and laminated plates are established by comparison with various exact and approximate solutions. The present Dynamic Relaxation method (DR) coupled with finite differences method shows a fairly good agreement with other analytical and numerical methods used in the verification scheme.It was found that: The convergence and accuracy of the DR solution is dependent on several factors including boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR large deflection program using uniform finite differences meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads. All the comparison results for simply supported (SS4) edge conditions showed that deflection is almost dependent on the direction of the applied load or the arrangement of the layers
Dynamic Relaxation (DR) method is presented for the analysis of geometrically linear laterally loaded, rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse shear deformations. A computer program has been compiled. The convergence and accuracy of the DR solutions of isotropic, orthotropic, and laminated plates for elastic small deflection response are established by comparison with different exact and approximate solutions. The present Dynamic Relaxation (DR) method shows a good agreement with other analytical and numerical methods used in the verification scheme. It was found that: The convergence and accuracy of the DR solution were dependent on several factors which include boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR small deflection program using uniform meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads in a fairly good accuracy.
First order orthotropic shear deformation equations for the linear elastic bending response of rectangular
plates are introduced. Their solution using a computer program in FORTRAN language based on finite differences
implementation of the Dynamic Relaxation (DR) method is outlined. The convergence and accuracy of the DR
solutions for elastic linear response of isotropic, orthotropic, and laminated plates are established by comparison with
various exact and approximate solutions. The present Dynamic Relaxation method (DR) coupled with Finite
Differences method (FD) shows a fairly good agreement with other analytical and numerical methods used in the
present study. It was found that the DR linear theory program using uniform meshes can be employed in the analysis
of different thicknesses and length to side ratios for isotropic, orthotropic and laminated fibrous plates under uniform
loads in a fairly good accuracy. These comparisons show that the type of mesh used (i.e. uniform or graded) is
responsible for the considerable variations in the mid – side and corner stress resultants. It is found that the
convergence of the DR solution depends on several factors including boundary conditions, meshes size, fictitious
densities and applied load. It is also found that the DR linear theory can be employed with less accuracy in the
analysis of moderately thick and flat isotropic, orthotropic or laminated plates under uniform loads. It is also found
that the deflection of the plate becomes of an acceptable value when the length to thickness ratio decreases. For
simply supported (SS1) edge conditions, all the comparison results confirmed that deflection depends on the direction
of the applied load and the arrangement of the layers.
Dynamic Relaxation (DR) method is presented for the analysis of geometrically linear laterally loaded, rectangular laminated plates. The analysis uses the Mindlin plate theory which accounts for transverse shear deformations. A computer program has been compiled. The convergence and accuracy of the DR solutions of isotropic, orthotropic, and laminated plates for elastic small deflection response are established by comparison with different exact and approximate solutions. The present Dynamic Relaxation (DR) method shows a good agreement with other analytical and numerical methods used in the verification scheme.
It was found that: The convergence and accuracy of the DR solution is dependent on several factors which include boundary conditions, mesh size and type, fictitious densities, damping coefficients, time increment and applied load. Also, the DR small deflection program using uniform meshes can be employed in the analysis of different thicknesses for isotropic, orthotropic or laminated plates under uniform loads in a fairly good accuracy.
Linear And Nonlinear Analytical Modeling of Laminated Composite Beams In Thre...researchinventy
The large current development of aerospace and automotive technologies is based on the use of composite materials which provide significant weight savings compared to their mechanical characteristics. Correct dimensioning of composite structures requires a thorough knowledge of their behavior in small as in large deflection.This work aims to simulate linear and nonlinear behavior of laminates composites under threepoint bending test. The used modelization is based on first-order shear deformation theory (FSDT), classical plate theory (CPT) and Von-Karman’s equations for large deflection. A differential equation of Riccati, describing the variation of the deflection depending on the load, was obtained. Hence, the results deduced show a good correlation with experimental curves
Characteristic orthogonal polynimial application to galerkin indirect variati...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Steady State Thermoelastic Problem in an Infinite Elastic Layer Weakened by a...inventionjournals
The paper is concerned with a thermoelastic behavior in an infinite elastic layer of finite thickness with a crack in it lying in the middle of the layer and parallel to the faces of the layer. The faces of the layer are maintained at constant temperature of different magnitude. The problem is a steady state thermoelastic problem. The layer surfaces are supposed to be acted on by symmetrically applied concentrated forces of magnitude with respect to the centre of the crack. The applied concentrated force may be compressive or tensile in nature. The problem is solved by using integral transform technique .The solution of the problem has been reduced to the solution of a Cauchy type singular integral equation, which requires numerical treatment. The stress-intensity factor and the crack opening displacement are determined and thermal effects on various subjects of physical interest are shown graphically
Acta Mater 57 (2009) 559 Investigation Of The Indentation Size EffectDierk Raabe
Investigation of the indentation size effect through the measurement of the geometrically necessary dislocations beneath small indents of different depths using EBSD tomography
A composite material can be defined as a combination of two or more materials that
gives better properties than those of the individual components used alone. In contrast to
metallic alloys, each material retains its separate chemical, physical, and mechanical
properties. The two constituents are reinforcement and a matrix. The main advantages of
composite materials are their high strength and stiffness combined with low density when
compared to classical materials. Micromechanical approach is found to be more suitable for
the analysis of composite materials because it studies the volume proportions of the
constituents for the desired lamina stiffness and strength.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Applications of layered theory for the analysis of flexible pavementseSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Nonlinear Viscoelastic Analysis of Laminated Composite Plates – A Multi Scale...rtme
Laminated composite plates are widely used in modern structures. Resins of composites are almost made of
polymers which show time dependent and and in some cases stress dependent behaviour. In this paper, a
laminated composite plate is analysed using a multiscale method. At first, material properties of a lamina is
obtained using an analytical micromechanical approach called simplified unit cell method (SUCM) and
then in macromechanical level, Generalized Differential Quadrature Method (GDQM) is used to analyse
laminated composite plate. Schapery's integral is used to model nonlinear viscoelastic behaviour of the
matrix. Prony series is considered to define the compliance of matrix. Micromechanical process includes
obtaining overall properties of the composite by SUCM. Both geometrical and material nonlinearity are
taken into account in order to multiscale analysis of laminated composite plate.
The effects of boundary conditions and lamination arrangements
(i.e. stacking sequence and orientation of a lamina) were found to be important
factors in determining a suitable exact, analytical or semi – analytical method
for analyzing buckling loads on laminated plates. It was also found that: as the
derivative order of shear deformation increases, the accuracy of stresses, strains,
buckling loads … etc. increases and it doesn't need shear correction factors.
EFFECT OF SURFACE ROUGHNESS ON CHARACTERISTICS OF SQUEEZE FILM BETWEEN POROUS...IAEME Publication
In investigation aims to analyse the effect of transverse surface roughness on the squeeze film performance between porous rectangular plates. The associated differential equation is stochastically averaged making use of stochastic averaging method of Christensen and Tonder for transverse surface roughness. The equation is solved with appropriate boundary conditions to obtain the pressure and consequentlythe load bearing. The graphical results suggest that the bearing suffers because of transverse surface roughness. However the situation is slightly better in the case of
negatively skew roughness. Further variance (-ve) makes the situation furtherimproved even if moderate values of porosity are involved
STATIC AND DYNAMIC ANALYSIS OF CENTER CRACKED FINITE PLATE SUBJECTED TO UNIFO...IAEME Publication
The study of crack behavior in a plate is a considerable importance in the design to avoid the failure. This paper deals with investigation of stress intensity factor, Von-Misesstress (ϬVon-mises),natural frequency, mode shape and the effect of excitation frequency on the finite center cracked plate subjected to uniform tensile loading depends on the assumptions of Linear Elastic Fracture Mechanics (LEFM) and plane strain problem.
First order shear deformation (FSDT) theory for laminated composite beams is used to study free vibration of
laminated composite beams, and finite element method (FEM) is employed to obtain numerical solution of the
governing differential equations. Free vibration analysis of laminated beams with rectangular cross – section for
various combinations of end conditions is studied. To verify the accuracy of the present method, the frequency
parameters are evaluated and compared with previous work available in the literature. The good agreement with
other available data demonstrates the capability and reliability of the finite element method and the adopted beam
model used.
Stability of Simply Supported Square Plate with Concentric CutoutIJMER
The finite element method is used to obtain the elastic buckling loads for simply supported isotropic square plate containing circular, square and rectangular cutouts. ANSYS finite element software had been used in the study. The applied inplane loads considered are uniaxial and biaxial compressions. In all the cases the load is distributed uniformly along the plate outer edges. The effects of the size and shape of concentric cutouts with different plate thickness ratios and having all-round simply supported boundary condition on the plate buckling strength have been considered in the analysis. It is found that cutouts have considerable influence on the buckling load factor k and the effect is larger when cutout ratios greater than 0.3 and for thickness ratio greater than 0.15.
ANALYTICAL BENDING ANALYSIS OF A CIRCULAR SANDWICH PLATE UNDER DISTRIBUTED LOADijmech
In this paper, bending analysis of a circular sandwich plate under distributed load with simply supported and clamped boundary conditions is investigated. First, the governing equations of the circular sandwich plate are obtained and they are solved using the Bessel functions. Then in order to validate the correctness of analytical results, numerical finite element method is used and its results are presented in the forms of
contours and graphs. The results indicate that under distributed load, maximum deflection happens at 0.3
of outside radius, away from centre, and minimum deflection occurs at the outer edge of the circular sandwich plate. The results from analytical and numerical methods are compared and it shows that analytical method provides an acceptable accuracy.
TRANSIENT ANALYSIS OF PIEZOLAMINATED COMPOSITE PLATES USING HSDTP singh
Piezoelectric materials have excellent sensing and actuating capabilities have made them the most practical smart materials to integrate with laminated structures. Integrated structure system can be called a smart structure because of its ability to perform self-diagnosis and quick adaption to environment changes. An analytical procedure has been developed in the work based on higher order shear deformation theory subjected to electromechanical loading for investigating transient characteristics of smart material plates. For analysis two displacement models are to be considered i.e., model-1 accounts for strain in thickness direction is zero whereas in model-2 in-plane displacements are expanded as cubic functions of the thickness coordinate. Navier’s technique has been adopted for obtaining solutions of anti-symmetric cross–ply and angle-ply laminates of both model-1 and model-2 with simply supported boundary conditions. For obtaining transient response of a laminated composite plate attached with piezoelectric layer Newmark’s method has been used. Effect of thickness coordinate of composite laminated plates attached with piezoelectric layer subjected to electromechanical loadings is studied.
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.
Analysis of Functionally Graded Material Plate under Transverse Load for Vari...IOSR Journals
Functionally gradient materials (FGM) are one of the most widely used materials in various
applications because of their adaptability to different situations by changing the material constituents as per the
requirement. Most structural components used in the field of engineering can be classified as beams, plates, or
shells for analysis purposes. In the present study the power law, sigmoid and exponential distribution is
considered for the volume fraction distributions of the functionally graded plates. The work includes parametric
studies performed by varying volume fraction distributions and boundary conditions. Also static analysis of
functionally gradient material plate is carried out by sigmoid law and verified with the published results. The
convergence study of the results is optimized by changing the mesh size and layer size. Power law and
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ANALYSIS FOR FREE VIBRATION OF LAMINATED COMPOSITE & SANDWICH PLATES WITH THE...IAEME Publication
This paper present the analytical solution for free vibration of Laminated composite and sandwich plates with the help of Euler-Lagrange Equation based on first order shear deformation theory to analytical formulations and solution to the natural frequency, analysis of simply supported
composite and sandwich plates on compared to obtained results, non dimensionalized fundamental frequencies.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
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Aerospace Engineering.
The determination of the stress distribution around Pin-Loaded Holes in Mechanically Fastened Joints is presented to the Engineering Department, University of Technology - Jamaica
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
Limit States Solution to CSCS Orthotropic Thin Rectangular Plate Carrying Tra...ijtsrd
The analysis of thin rectangular orthotropic plate with two opposite edges clamped and the other two opposite edges simply supported CSCS , carrying transverse loads was investigated in this study. The Ritz total potential energy functional was used. The minimization of the total potential energy functional produces the expression for the coefficient of deflection. The coefficient of deflection was used to obtain equation for the maximum lateral load of an orthotropic thin rectangular plate based on allowable deflection. Also, equation for the maximum lateral load of an orthotropic thin rectangular plate based on allowable stress was developed.Developed stiffness coefficients were substituted in the lateral load equations to obtain the maximum lateral load values for a CSCS plate. Numerical examples using permissible deflection of 10mm and yield strength of 250MPa, plate thickness varying from 5mm to 12.5 mm with 0.5mm intervals were done to determine the maximum lateral loads corresponding to an orthotropic thin rectangular CSCS plate carrying transverse loads when n1 = Ey Ex = 0.7 and n2 = G Ex = 0.41 for aspect ratios b a of 1.0, 1.25 and 1.50. Bertram D. I. | Okere C. E. | Ibearugbulem O. M. | Nwokorobia G. C. "Limit States Solution to CSCS Orthotropic Thin Rectangular Plate Carrying Transverse Loads" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-6 , October 2021, URL: https://www.ijtsrd.com/papers/ijtsrd47566.pdf Paper URL : https://www.ijtsrd.com/engineering/civil-engineering/47566/limit-states-solution-to-cscs-orthotropic-thin-rectangular-plate-carrying-transverse-loads/bertram-d-i
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
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Bob Boule
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Gopinath Rebala
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LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
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PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
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The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
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Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
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Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
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Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
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Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
DevOps and Testing slides at DASA ConnectKari Kakkonen
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4. Demo
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Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
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Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...
J012236977
1. IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE)
e-ISSN: 2278-1684,p-ISSN: 2320-334X, Volume 12, Issue 2 Ver. III (Mar - Apr. 2015), PP 69-77
www.iosrjournals.org
DOI: 10.9790/1684-12236977 www.iosrjournals.org 69 | Page
The Thick Orthotropic Plates Analysis Methods, Part I: A Review
Parham Mohajerani
Lecturer, Department of Civil Engineering, Payame Noor University, Esfahan, Iran
Abstract: As usage of plates, especially thick plates, are increased in structural and mechanical projects, the
need for more precise plate analysis method increased, too. Designers look for a plate analysis method which
can result in an exact stress-strain relationship with consideration of thickness effects, shear deformation and
compressive deformation of plate across the thickness. This paper reviews the process of thick plate analysis
method developments by describing a number of main plate analytical methods. Weak and strong points of each
method are explained for each method.
Keywords: Exact 3D solution, Orthotropy,State Space method, Thick plate, Plate analysis
I. Introduction
The Plates are straight and plane surface structures whose thickness is slight compared to other
dimensions geometrically. The Classical Plate Theory (CLPT) limitation for the solution of flexural problems
associated with composite laminates have been investigate by several research workers [1-4]. In many
researches, it has been shown that the effect of transvers shear deformation is important. Mindlin’s plate theory
[5] which allows for this type of defamation, has been used as a basis for obtaining more accurate solutions [6].
Classical Plate theory is based on its use of conventional isotropic thin plate theory and transverse shear
deformation and rotatory inertia effects neglected in the laminate. Hence, this analytical method yields only
average through thickness values for the in-plane stress and gives inadequate values about the essential inter-
laminar shear stress, which can be the cause for delamination fact. Therefore, researchers try to use three-
dimensional equilibrium that takes into account three-dimensional variation of stresses and strains. This idea
helps them to provide more accurate response of laminated composite structures [7, 8].
Method of Initial Function (MIF) [9], Finite Strip Method [10], State Space Method [11, 12] and Finite
Element Method [13] are four main methods have been used to solve composite laminate plates structures.
Method of Initial Function (MIF) is one of the most accurate method in elastic analysis of orthotropic
composite material which proposed by VasiliĭZakharovichVlasov (1955). V.Z, Vlasov offered his method to
solve problems in theory of elasticity. Bahar, Das and Rao introduced the state space and used matrix method in
solving progress of MIF [11, 12].
In addition, plate thickness can affect stress-strain results and displacement outputs. Classical Plate
theory and some other solutions (i.e. Reissner and Ambartsumyan) have not been mentioned the effect of z
variation in their solution. For instance, Ressiner, Vlasov and Mindlin assumed different functions for
compressive deformation, but none of them has no relationship to coordinate z. In 1989, Sundara Raja Iyengar
and Pandya used a sixth-order governing equation to analyze uniformly loaded simply supported square plates
for various thickness and material properties [14] and they inserted thickness factor in their solution.
Exact 3-D solution is the most accurate method based on state space method which is involved with
matrix method. The problem of bending of simply supported thick orthotropic rectangular plates and laminates
was solved by Rao and Sirinivas[2]. Then, Fan and Ye (1990) have derived solutions in the form of double
Fourier series. Jiarang Fan used State Space method for solving thick orthotropic structures [15] with different
boundary conditions. Wu and Wardenier in 1998 proposed exact elastic solution for anisotropic thick
rectangular plates with four edges simply supported [16].
Fig. 1 Load & boundary conditions and general geometry of problem.
2. The Thick Orthotropic plates analysis methods, Part I: A Review
DOI: 10.9790/1684-12236977 www.iosrjournals.org 70 | Page
II. Elastic Solutions for Anisotropic Simply supported Rectangular Plates
There are different methods for elastic analysis of anisotropic plates. In some methods, compressive
deformation is neglected. In some others, researchers got this deformation linear or cubic function. For example,
Vlasov assumed this deformation has a cubic function and Hencky in 1947 assumed a linear function for
compressive deformation.
Reissner got compressive deformation is a cubic function across the thickness. Mindlin assumed that
function of compressive deformation is sine function. However, the deflection in z direction has no relationship
to coordinate z.
Method of initial function (MIF) was one of the first methods that used to investigate the bending
problem of rectangular isotropic material by Sundara Raja Iyengar. Bahar, Das and Rao used the state space and
matrix method to the MIF [11]. Iyengar and pandya were the first researchers who mentioned this idea that
compressive deformation in anisotropic material related to z coordinate and Pagano [17] had the same idea
about the effect of z coordinate in compressive deformation value through the thickness. However, his
calculation was limited. Wu and Wardenier (1997) removed the limitation in Pagano’s solution for anisotropic
rectangular simply supported plate and they provided six-order differential equation for estimating of transverse
displacement w in simply supported rectangular tick plate.
2.1 Ambartsumyan’s Theory
Ambartsumyan (1969) introduced quadratic function for shear stress variation across the thickness.
However, in his theory, compressive deformation was z invariant. He presented an elastic solution for bending
of hinged orthotropic rectangular plate. Based on Eq. (1) presented in his analytical method, the compressive
deformation is not z dependent. fmnin Eq. (1) is an unknown coefficient and it can be calculated based on Eq. (2).
Ambartsumyan, based on Navier solution (1820), used Eq. (3) to represent the load function which has a double
Fourier series function. amn as a load function factor for uniformly distributed load was proposed in form of Eq.
(4).
Due to z invariant of fmn and amn, the transverse deformation is constant based on Ambartsumyan
solution.
ω = ∑ ∑ fmn sin
mπx
a
∞
n=1
∞
m=1
sin
nπy
b
(1)
fmn =
∆1mn
∆0mn
amn (2)
F(x, y) = ∑ ∑ amn sin
mπx
a
∞
n=1
∞
m=1
sin
nπy
b
(3)
𝑎 𝑚𝑛 =
4
𝑎𝑏
∫ ∫ 𝑓(𝑥, 𝑦)
𝑏
0
𝑎
0
𝑠𝑖𝑛
𝑚𝜋𝑥
𝑎
𝑠𝑖𝑛
𝑛𝜋𝑦
𝑏
𝑑𝑥 𝑑𝑦 (4)
(5)
∆0𝑚𝑛=
ℎ3
12
[𝐷11
𝜋4
𝑚4
𝑎4
+ 2(𝐷12 + 2𝐷66)
𝜋4
𝑚4
𝑛2
𝑎2 𝑏2
+ 𝐷22
𝜋4
𝑛4
𝑏4
]
+
ℎ2
10
[(𝐷11
𝜋2
𝑚2
𝑎2
+ 𝐷66
𝜋2
𝑛2
𝑏2
) (𝐷22
𝜋2
𝑛2
𝑏2
+ 𝐷66
𝜋2
𝑚2
𝑎2
)
− (𝐷12 + 𝐷66) 2 𝜋4
𝑚2
𝑛2
𝑎2 𝑏2
] (𝑎44
𝜋2
𝑚2
𝑎2
+ 𝑎55
𝜋2
𝑛2
𝑏2
)
3. The Thick Orthotropic plates analysis methods, Part I: A Review
DOI: 10.9790/1684-12236977 www.iosrjournals.org 71 | Page
(6)
∆1𝑚𝑛=
12
ℎ3
{
ℎ6
144
+
ℎ5
120
[𝑎55 (𝐷11
𝜋2
𝑚2
𝑎2
+ 𝐷66
𝜋2
𝑛2
𝑏2
) + 𝑎44 (𝐷22
𝜋2
𝑛2
𝑏2
+ 𝐷66
𝜋2
𝑚2
𝑎2
)]
−
ℎ4
100
𝑎44 𝑎55 [(𝐷12 + 𝐷66)
𝜋4
𝑚2
𝑛2
𝑎2 𝑏2
− (𝐷11
𝜋2
𝑚2
𝑎2
+ 𝐷66
𝜋2
𝑛2
𝑏2
)
× (𝐷22
𝜋2
𝑛2
𝑏2
+ 𝐷66
𝜋2
𝑚2
𝑎2
)]}
+
ℎ2
10
{𝐴1 [
ℎ3
12
𝜋2
𝑚2
𝑎2
+ 𝑎44
ℎ2
10
𝜋2
𝑚2
𝑎2
(𝐷22
𝜋2
𝑛2
𝑏2
+ 𝐷66
𝜋2
𝑚2
𝑎2
) − 𝑎55
ℎ2
10
(𝐷12 + 𝐷66)
𝜋4
𝑚2
𝑛2
𝑎2 𝑏2
]
+ 𝐴2[
ℎ3
12
𝜋2
𝑛2
𝑏2
+ 𝑎55
ℎ2
10
𝜋2
𝑛2
𝑏2
(𝐷11
𝜋2
𝑚2
𝑎2
+ 𝐷66
𝜋2
𝑛2
𝑏2
) − 𝑎44
ℎ2
10
(𝐷12 + 𝐷66)
𝜋4
𝑚2
𝑛2
𝑎2 𝑏2
}
where ω is transverse deformation, fmn is transverse deformation coefficient, a is rectangular length in x
direction and b denotes rectangular length in y direction. ∆1mn, ∆0mn are related to transverse deformation
coefficient factors. F(x, y) is load function and amn denoting its factor. amnfor uniform distributed load of q is
equal to
16𝑞0
𝑚.𝑛.𝜋2 .
Fig. 2 Ambartsumyan’s solution plate geometry.
2.2 Kirchhoff-Love Theory
Kirchhoff-Love theory is also popular as “Classic Plate Theory “. Gustavo Kirchhoff has worked on
stress and deformation determination in thin plates. In 1888, Augustus Edward Hough Love, the British
mathematician, worked on theory of elasticity. He used Kirchhoff assumptions and tried to develop his idea.
Kirchhoff thin plate theory was based on Euler-Bernoulli beam theory. He assumed, straight line
normal to the mid-surface remain straight after deformation. In addition, he assumed straight line normal to the
mid-surface remain normal to the mid-surface after bending deformation. It means that effect of shear stress was
neglected in his theory. He also assumed the thickness of plate does not change during a deformation [18].
Three dimensional solutions proved that, this assumption is wrong.
Kirchhoff and love changed a 3-D plate into 2-D plate of middle surface plane. This assumption makes
elastic plate analysis fallowed by the plane stress solution of plate bending problem. However, it leads to
removing of z in their solution. Their solution is not z variant for deformation across the plate thickness.
D∇4
w = q(x, y) (7)
∇4
≡ (
∂4
∂x4
+ 2
∂4
∂x2 ∂y2
+
∂4
∂y4
) (8)
where w is transverse deflection; ∇ is the biharmonic differential operator (∇2
=
𝜕2
𝜕𝑥2 +
𝜕2
𝜕𝑦2 rectangular co-
ordinates); D = Eh3
/12(1 - 𝜐2
), the flexural rigidity; E is Young's modulus; h is plate thickness; 𝑣 is Poisson's
ratio. Exclude such complicating effects as orthotropy, in-plane forces, variable thickness, large deflections,
shear deformation and rotary inertia, and non-homogeneity [19].
4. The Thick Orthotropic plates analysis methods, Part I: A Review
DOI: 10.9790/1684-12236977 www.iosrjournals.org 72 | Page
2.3 Mindlin & Reissner Plate Theories
Raymond Mindlin and Eric Reissner extended the Kirchhoff-love plate theory. They tried to make a
better approximation for solving bending of elastic plates. In 1945, Reissner proposed his idea that shear
deformation can effect on the bending of elastic plates. Reissner assumed that the bending stress is linear while
shear stress is parabolic through the thickness. It means that the normal to the mid-surface remains straight but
not necessarily perpendicular to it.
Mindlin assumed that compressive deformation across the plate thickness has linear variation. However
in his theory, the plate thickness does not change after deformation [5]. Mindlin ignored the normal stress
through the thickness and this assumption leads to plane-stress condition problem. In contrast, Reissner accounts
this normal stress in his plate theory [20]. Both Mindlin and Reissner theories independently proposed plate
theories that incorporate the effect of transvers shear deformation for analysing “thick plate” (Wang et al.,
2001). As it has mentioned before, Reissner is the first researcher who proposed effect of transverse shear
deformation on the bending of elastic plates. In Eqs. (9) and (10), the displacement in n and s directions (n= x
direction, s= y direction) contain of shear term. These two formulas indicate that, due to the effect of shear
deformation, normal and tangential line elements in middle surface do not remain perpendicular to the linear
element which was before deformation perpendicular to the middle surface [21].
∂w
∂n
−
12(1 + υ)
5hE
Vn = −U̅n′ (9)
∂w
∂s
−
12(1 + υ)
5hE
Vs = −U̅s′ (10)
where:
𝑢 𝑛′̅̅̅̅ : Displacement in direction n [22]
𝑢 𝑠′̅̅̅̅ : Displacement in direction s (y)
𝑤 : Displacement component normal to the plane of the plate
Reissner proposed this idea that w is biharmonic function. It means, he assumed w has a forth order
partial differentiation function.
2.4 Viktor Valentinovich Vlasov Method
V.V. Vlasov used Method of Initial Function (MIF) to solve displacement function for simply
supported plate [23]. He analyzed plate bending behavior based on MIF for homogenous and non-homogenous
material. As it has mentioned in Eqs. (11) and (12), w is the displacement through the thickness and it has
biharmonic function. However, he proved this equation based on z= ± h/2. By setting z= ±h/2, the results
shows, the displacement across the thickness is not z variant. He proposed Eq. (11) to calculate w in the case of
homogenous plate bending condition:
w = c1 + c2x + c3x2
+ c4x3
+
ν
1 − ν
z2(c3 + 3c4x) (11)
and Eq. (12) for non-homogenous case defined as:
(12)
w =
q
24D
{x(a3
− 2ax2
+ x3) +
3
10
8 − 3υ
1 − υ
h2
x(a − x) + z2
[
3(5 − 3υ)
10(1 − υ)
h2
+
6υ
1 − υ
x(x − a) −
1 + υ
1 − υ
z2
]
where:
z = ±
ℎ
2
h : Plate thickness
a : a ×a rectangular plate
In Eqs. (11) and (12), z can be set as h/2 or –h/2. This means that, V. V. Vlasov believed that vertical
displacement across the thickness is constant.in both cases (h/2 or –h/2), because of even power of z variable,
the value of w at top and bottom of plate at specific x and y are similar. Similarity in the value of top and
bottom surface is not acceptable because Wu and Wardenier proved that displacement of top is more than
bottom for simply supported anisotropic thick plates.
Fig. 3 Vlasov solution geometry.
5. The Thick Orthotropic plates analysis methods, Part I: A Review
DOI: 10.9790/1684-12236977 www.iosrjournals.org 73 | Page
2.5 Pagano Solution
Pagano (1969) worked on exact elastic response of rectangular, pinned edge laminates consisting of
any number of orthotropic or isotropic layers [24]. On the other hand, he has presented further evidence
regarding the range of validity and limitation of CLPT. He came up with six-order partial differentiation (Eq.
13), which is z variant. He proved his idea by using Cardano’s mathematical method to solve the differential
equation. However, Pagano assumed H factor in Cardano’s method is always negative for all material (Eq. 16).
This assumption is incorrect and Wu and Wardenier (1997) proved H can be any real number. The sign of H
related to the combination of geometric properties and material properties.
Fig. 4 Pagano’s solution geometry.
−𝐴𝑠6
+ 𝐵𝑠4
+ 𝐶𝑠2
+ 𝐷 = 0 (13)
𝛾3
+ 𝑑𝛾 + 𝑓 = 0 (14)
𝛾 = 𝑠2
−
𝐵
3𝐴
(15)
𝐻 =
𝑓2
4
+
𝑑3
27
(16)
where:
S:
∂w
∂z
A, B, C and D: coefficients that consist of elastic coefficient and load
f, d: Cardano’s method coefficient
This equations is true for simply supported rectangular plates [24].
Another idea which Pagano insisted on is related to convergence of the elasticity solution to classical
plate theory. He got this general idea that convergence of the elasticity solution to CLPT is more rapid for the
stress components than the plate deflection. Pagano believed that this observation could be a good consideration
in selecting the form of plate theory required in the solution of specific boundary value problem [24].
2.6 K.T.Sundara Raja Iyengar and S.K.Pandya Solution
K.T.Sundara Raja Iyengar and S.K.Pandya (1983) worked on problem of analyzing orthotropic
rectangular plate based on initial function method. As it has mentioned, Vlasov introduced this method of initial
function. In this method, there is not any assumption that was made regarding the distribution of stress and
displacement. In 1975, Bahar proposed “A State Space approach to elasticity “to solve the initial function
results. Iyengar and Pandya used this approach in order to formulate the general problem of orthotropic
rectangular thick plate analysis [14].
In addition, they used Navier solution for uniformly distributed load on a simply supported square
plate. In State Space method, the value of stress and displacement are related to the value of stress and
displacement in a plate where z = 0 (Eq. 20). Pandya used the equilibrium equations as:
(17)
σx = C11εx + C12εy + C13εz
6. The Thick Orthotropic plates analysis methods, Part I: A Review
DOI: 10.9790/1684-12236977 www.iosrjournals.org 74 | Page
σy = C12εx + C22εy + C23εz
σz = C13εx + C23εy + C33εz
τyz = Gyzγyz
τxz = Gxzγxz
τxy = Gxyγxy
By elimination of σx ,σy,τxy and the strains and using the strain-displacement and equilibrium equations we can
get :
∂
∂z
{
U
V
Z
X
Y
W}
=
[
0
0
0
−C2α2
− C6β2
−(C3 + C6)αβ
C1α
0
0
0
−(C3 + C6)αβ
−C6α2
− C4β2
C5β
0
0
0
C1α
C5β
C10
a11
0
−α
0
0
0
0
a22
−β
0
0
0
−α
−β
0
0
0
0 ] {
U
V
Z
X
Y
W}
(18)
∂
∂z
{S} = [D]{S} (19)
{S} = [e[D]z
]{S0} (20)
[L] = e[D]z
= [I] + z[D] +
z2
2
[D]2
+ ⋯ (21)
where:
{S}: state vector
{S0}: The value of state vector on initial plane (z=0)
[L]: operator matrix of mapping functions
[D]: System Matrix
C1...C10 :Mechanical constants [15]
U, V and W : displacement in x, y and z direction
As shown in Fig. 5, the results of MIF and Reissner’s theory are the same in case of 𝜎𝑧𝑧 across the
thickness. They also proved that the maximum deflection variation across the plate thickness is not constant.
Fig.6 clearly reveals that the top loaded surface of plate has higher deflection compared to the lower one.
Fig. 5 Variation of 𝜎𝑧 across thethickness [14].
(
𝑎
ℎ
=2.5, 5,10,20,40 and 80 ;
𝐸 𝑥
𝐸 𝑦
= 1, 3, 10 and 40; 𝐸𝑧 = 𝐸 𝑌; 𝐺 𝑦𝑧 = 0.5 × 𝐸 𝑦; 𝐺𝑥𝑧 =
𝐺𝑥𝑦 = 0.6 × 𝐸 𝑦; 𝜇 𝑥𝑦 = 𝜇 𝑥𝑧 = 𝜇 𝑦𝑧 = 0.25 )
7. The Thick Orthotropic plates analysis methods, Part I: A Review
DOI: 10.9790/1684-12236977 www.iosrjournals.org 75 | Page
Fig. 6Variation of 𝑤 𝑚𝑎𝑥 across the thickness [14].
(
𝑎
ℎ
= 2.5;
𝐸 𝑥
𝐸 𝑦
= 1, 3, 10 and 40; 𝐸𝑧 = 𝐸 𝑌; 𝐺 𝑦𝑧 = 0.5 × 𝐸 𝑦; 𝐺𝑥𝑧 = 𝐺𝑥𝑦 = 0.6 × 𝐸 𝑦;
𝜇 𝑥𝑦 = 𝜇 𝑥𝑧 = 𝜇 𝑦𝑧 = 0.25 )
2.7 Wu and Wardenier
In 1997, they used state space method to solve the problem of anisotropic thick rectangular plates in
bending (Fig. 7). Wu and Wardenier used State Space method for getting the result for simply supported thick
rectangular plate. They found a graph which shows the variation of displacement across the thickness of plate
(Fig. 8). They compared their own calculations results with Ambartsumyan and Reissner’s theories results [25].
Fig. 7 Plate Geometry[16].
Fig. 8 Maximum transvers displacement across the thickness of the plate (h/a=0.2 loaded
on its upper surface (z=0) by uniform normal loading q0 [16].
8. The Thick Orthotropic plates analysis methods, Part I: A Review
DOI: 10.9790/1684-12236977 www.iosrjournals.org 76 | Page
d6Wmn
d6 + A0
d4Wmn
d4 + B0
d2Wmn
d2 + C0Wmn = 0 (22)
They proved that the variation of maximum transvers displacement across the thickness is nonlinear and they
introduced a six-order differential equation for transverse displacement (w) by using State Space method.
2.8 Fan’s State Space Solution
In early 90s, Jia-rang Fan believed that traditional theories of thick plate analysis are established based
on assumptions which are limited those methods. He believed that some elastic constant had been missed in
traditional plate analysis methods. Thus, by using State Space method he solved the exact behavior of thick
rectangular plate with simply supported edges. He introduced Eqs. (23-25) as a system matrixes of his solution
based on Eq. 19 which has differently introduced by K.T.Sundara Raja Iyengar and S.K.Pandyain 1983.
(23)
d
dz
[Umn(z)Vmn(z)Zmn(z)Xmn(z)Ymn(z)Wmn(z)]T
= [
0 Amn
Bmn 0
] [ Umn(z)Vmn(z)Zmn(z)Xmn(z)Ymn(z)Wmn(z)]T
Amn = [
C8 0 −ξ
0 C9 −η
ξ η 0
] (24)
Bmn = [
C2ξ2
+ C6η2 (C3 + C6)ξη C1ξ
(C3 + C6)ξη C6ξ2
+ C4η2
C5η
−C1ξ −C5η C10
] (25)
where:
ξ =
mπ
a
η =
nπ
b
Eq.23 shows that stresses and displacements are directly related to coordinate z in State Space solution by Fan
and none of elastic constant has been missed in his solution.
III. Conclusion
Many well-known researchers have done researches on elastic bending plate analysis. They all tried to
get the real stress-strain relation from their methods. Some of them, such as Kirchhoff and Love, introduced
their solution for thin plate bending analysis and they remove plate thickness from their analysis procedure.
Then, Ambartsumyan believed that z coordinate should be involved in thick plate analysis and it should be
involved in displacement across the thickness. However, the way he used thickness factor (i.e. z coordinate) in
his formulas, the z factor removed due to even power of z and the two specific value of z, ±h/2. Thus
displacement across the thick ness is not z variant and it is become constant.
Reissner introduced shear deformation effect on bending of elastic plate and Mindlin assumed linear
compressive deformation across the plate thickness.
Pagano, Iyengar and Pandya were three researchers who mention z factor in their solutions. Pagano
come up with six-order differential equation which has developed by Wu and Wardenier later. Iyengar and
Pandya used method of initial function and improved state space method. In 90s, Fan used state space method to
find the exact solution of elastic bending of thick plate. He used this method to develop the exact stress-strain
relations in bending problem of thick orthotropic plate. He used this method to solve bending behavior of thick
plate with different boundary conditions, such as simply supported rectangular plate and four edges clamped
boundaries. He did not touch the problem of elastic bending of thick orthotropic plate with symmetric clamped-
free edges. Author will developed State Space solution for this specific boundary condition based on this review
paper.
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