1) The document analyzes stresses and displacements around cylindrical cavities in an elastic medium caused by moving step loads on the cavity surface.
2) Governing equations are derived using potential functions and Fourier series in cylindrical coordinates. The equations are reduced to Helmholtz equations for the potentials.
3) Solutions are obtained for axisymmetric and non-axisymmetric step loads moving parallel to the cavity axis. Numerical solutions provide stress and displacement values.
This document summarizes key concepts about wave forces on slender cylinders from Chapter 12 of the textbook "Offshore Hydromechanics". It discusses:
1) The two main inertia force components - the pressure gradient force (Fx1) and disturbance force (Fx2) - which together make up the total inertia force (FI).
2) The theoretical and experimental values for the inertia coefficient (CM), which is usually between 1-2 rather than the theoretical value of 2.
3) How the disturbance force coefficient (Ca) is often less than 1 due to flow disturbances, and how it represents force per unit acceleration rather than a physical mass of fluid.
This chapter discusses rigid body dynamics and ship motions. It introduces coordinate systems used to define ship motions, including translations and rotations of the ship's center of gravity. Key concepts covered include frequency of encounter, definitions of various ship motions like surge, sway, heave, roll, pitch and yaw. The chapter also discusses determining absolute and relative vertical motions of points on a ship's structure through superposition of heave, roll and pitch motions. As an example, ship motions are related to a simple single linear mass-spring system.
Analysis of the Interaction between A Fluid and A Circular Pile Using the Fra...IJERA Editor
The purpose of this research is to study the interaction between a fluid and a circular pile, located downstream
from a fan-shaped dam, through the fractional Navier-Stokes equations, and in particular, its approximation to
the boundary layer. The flow region is divided into zones according to the vorticity transport theory of
turbulence. First, we consider the limit of the spatial occupancy index close to 1. Then, a stream function is
introduced, and for the potential zone, we consider a complex potential, using the inverse distances on a circle.
In the other limit, when the spatial occupation index approaches 0, we consider the equations of the boundary
layer in the limit of fully developed turbulence. Next, for the last approaches, a new stream function and
velocities in their radial and polar components are obtained. We also find the asymmetry of the pressure
distribution around the pile, based on the viscosity and considering that the pressure drag force and the friction
coefficients are proportional to the inverse of the Reynolds number. We conclude that D'Alembert's paradox and
Thomson's theorem has been resolved. For applications, in the case of the turbulent wake, we are interested both
in the orientation given by the pile symmetry axis and its extension. The criterion that should be satisfied is: the
diameter of the pile, on the border of inequality, must be located as proportional average between the length of
the turbulent wake and twice the characteristic length associated with the dam, whose aspect ratio, in turn, to the
pile diameter, determines the contraction factor.
LES-DQMOM based Studies on Reacting and Non-reacting Jets in Supersonic Cross...Samsung Techwin
This document summarizes a presentation given at the 50th AIAA Aerospace Science Meeting on large eddy simulation (LES) studies of reacting and non-reacting transverse jets in supersonic crossflow. The presentation discusses the numerical methodology used, including the compressible flow solver and direct quadrature method of moments (DQMOM) combustion model. Results are presented for non-reacting and reacting jet in supersonic crossflow cases, including comparisons to experimental data. Key flow features like shock structures and vortical structures are analyzed.
1) Streamwise vortices play an important role in sustaining wall turbulence by regenerating streaks through the lift-up effect.
2) In turbulent plane Couette flow at low Reynolds numbers, streamwise vortices that span the entire gap between plates have been observed.
3) The document proposes a two-step Galerkin projection method to derive a low-order model that can illustrate the dynamics and generation mechanism of these streamwise vortices, in a way that is analogous to what is observed in turbulent boundary layers.
The document summarizes a study on the effect of jet configuration on transverse jet mixing. Direct numerical simulations were performed to analyze the effect of jet velocity profile and exit shape. Results show that a parabolic velocity profile enhances mixing over a top-hat profile due to slower vortex breakdown. For exit shape, a circular jet exhibits the most efficient mixing while triangular jets display two counter-rotating vortex pairs that increase entrainment and mixing.
This document describes a study on designing a 2D ocean wave maker. It presents the wavemaker theory for plane waves produced by a paddle. The key points are:
1) The governing equations and boundary conditions for linear water wave theory are described, including the Laplace equation, kinematic and dynamic free surface boundary conditions, and bottom and lateral boundary conditions.
2) The boundary value problem is solved using separation of variables. This results in a dispersion relationship for progressive waves and an equation relating wave numbers of standing waves to the wavemaker frequency.
3) The velocity potential solution is a superposition of progressive and decaying standing waves. The coefficients are determined by satisfying the lateral boundary condition at the wavemaker.
This document summarizes key concepts about wave forces on slender cylinders from Chapter 12 of the textbook "Offshore Hydromechanics". It discusses:
1) The two main inertia force components - the pressure gradient force (Fx1) and disturbance force (Fx2) - which together make up the total inertia force (FI).
2) The theoretical and experimental values for the inertia coefficient (CM), which is usually between 1-2 rather than the theoretical value of 2.
3) How the disturbance force coefficient (Ca) is often less than 1 due to flow disturbances, and how it represents force per unit acceleration rather than a physical mass of fluid.
This chapter discusses rigid body dynamics and ship motions. It introduces coordinate systems used to define ship motions, including translations and rotations of the ship's center of gravity. Key concepts covered include frequency of encounter, definitions of various ship motions like surge, sway, heave, roll, pitch and yaw. The chapter also discusses determining absolute and relative vertical motions of points on a ship's structure through superposition of heave, roll and pitch motions. As an example, ship motions are related to a simple single linear mass-spring system.
Analysis of the Interaction between A Fluid and A Circular Pile Using the Fra...IJERA Editor
The purpose of this research is to study the interaction between a fluid and a circular pile, located downstream
from a fan-shaped dam, through the fractional Navier-Stokes equations, and in particular, its approximation to
the boundary layer. The flow region is divided into zones according to the vorticity transport theory of
turbulence. First, we consider the limit of the spatial occupancy index close to 1. Then, a stream function is
introduced, and for the potential zone, we consider a complex potential, using the inverse distances on a circle.
In the other limit, when the spatial occupation index approaches 0, we consider the equations of the boundary
layer in the limit of fully developed turbulence. Next, for the last approaches, a new stream function and
velocities in their radial and polar components are obtained. We also find the asymmetry of the pressure
distribution around the pile, based on the viscosity and considering that the pressure drag force and the friction
coefficients are proportional to the inverse of the Reynolds number. We conclude that D'Alembert's paradox and
Thomson's theorem has been resolved. For applications, in the case of the turbulent wake, we are interested both
in the orientation given by the pile symmetry axis and its extension. The criterion that should be satisfied is: the
diameter of the pile, on the border of inequality, must be located as proportional average between the length of
the turbulent wake and twice the characteristic length associated with the dam, whose aspect ratio, in turn, to the
pile diameter, determines the contraction factor.
LES-DQMOM based Studies on Reacting and Non-reacting Jets in Supersonic Cross...Samsung Techwin
This document summarizes a presentation given at the 50th AIAA Aerospace Science Meeting on large eddy simulation (LES) studies of reacting and non-reacting transverse jets in supersonic crossflow. The presentation discusses the numerical methodology used, including the compressible flow solver and direct quadrature method of moments (DQMOM) combustion model. Results are presented for non-reacting and reacting jet in supersonic crossflow cases, including comparisons to experimental data. Key flow features like shock structures and vortical structures are analyzed.
1) Streamwise vortices play an important role in sustaining wall turbulence by regenerating streaks through the lift-up effect.
2) In turbulent plane Couette flow at low Reynolds numbers, streamwise vortices that span the entire gap between plates have been observed.
3) The document proposes a two-step Galerkin projection method to derive a low-order model that can illustrate the dynamics and generation mechanism of these streamwise vortices, in a way that is analogous to what is observed in turbulent boundary layers.
The document summarizes a study on the effect of jet configuration on transverse jet mixing. Direct numerical simulations were performed to analyze the effect of jet velocity profile and exit shape. Results show that a parabolic velocity profile enhances mixing over a top-hat profile due to slower vortex breakdown. For exit shape, a circular jet exhibits the most efficient mixing while triangular jets display two counter-rotating vortex pairs that increase entrainment and mixing.
This document describes a study on designing a 2D ocean wave maker. It presents the wavemaker theory for plane waves produced by a paddle. The key points are:
1) The governing equations and boundary conditions for linear water wave theory are described, including the Laplace equation, kinematic and dynamic free surface boundary conditions, and bottom and lateral boundary conditions.
2) The boundary value problem is solved using separation of variables. This results in a dispersion relationship for progressive waves and an equation relating wave numbers of standing waves to the wavemaker frequency.
3) The velocity potential solution is a superposition of progressive and decaying standing waves. The coefficients are determined by satisfying the lateral boundary condition at the wavemaker.
This document introduces new associated curves called k-principal direction curves and kN slant helices for spatial curves. It defines k-principal direction curves as integral curves of the k-th principle normal vector of the curve. A curve is a kN slant helix if its k-principal direction curve has constant geodesic curvature. The document establishes properties of the Frenet frame and curvature formulas for k-principal direction curves. It explores using these new curves to characterize different types of spatial curves.
Nurek rockfill dam (300 m). Problem of seismic safety of dam (4 p.)Yury Lyapichev
Serious problem of dam seismic safety was tried to be solved by reinforced concrete belt-elements incorporated in the upper part of upstream zone & clay core. But due to large construction settlements of this zone & core this solution was useless & expensive
This document provides an introduction to static floating stability of offshore structures. It defines key terms like center of buoyancy, center of gravity, metacenter and righting moment. It describes the principles of vertical and rotational equilibrium for floating bodies. As an example, it then calculates the static stability of a rectangular pontoon undergoing a lifting operation, finding the maximum angle of heel before the structure reaches equilibrium.
This document summarizes a research paper that develops a mathematical model for analyzing the three-dimensional shape of a long twisted rod hanging under gravity, such as a pipeline being laid from a barge. The model uses the geometrically exact theory of linear elastic rods and formulates the problem as a boundary value problem that is solved using matched asymptotic expansions. The truncated analytical solution is compared to results from a numerical scheme and shows good agreement. The method is then applied to consider the near-catenary shape of a clamped pipeline during the laying process.
Simulation of segregated flow over the 2 d cylinder using star ccm+Burak Turhan
In this thesis numerical simulation for classical case of flow over a cylinder is accomplished for 2D models using commercial CFD code Star CCM+ with k-ϵ model approach. The results are validated by comparing the Drag coefficients to the previously published data. The simulation is carried out to for Reynolds number 3900 to investigate the turbulence modeling on separation from curved surfaces of two different sizes of a circular cylinder, a cylinder with triangular cross section and a rectangular cross section. Investigation of different turbulence models and Mesh convergence is carried out.
The investigation of the turbulence model of the circular cylinder is carried out by the drag coefficient obtained by four different turbulence models such as K-Epsilon Turbulence, K-Omega Turbulence, Reynolds Stress Turbulence and Spalart-Allmaras Turbulence. Drag coefficient found out by different turbulence model is compared with the experimental value of a previously published data. The Mesh Convergence have been carried out by implementing different base mesh size in a decreasing order and the convergence is obtained when the drag coefficient becomes constant
The document discusses slope mass rating (SMR) and its use in assessing the stability of rock slopes. SMR is calculated based on the basic rock mass rating (RMR) minus adjustment factors (F1, F2, F3) that account for discontinuity orientation plus an additional factor (F4) depending on excavation method. SMR values are used to classify slope stability into five classes, with recommended support measures depending on the class such as bolting, shotcrete, or retaining walls. The document also discusses factors that can affect slope stability and adaptations made to the SMR system for use in different regions.
Chapter 1 - rotational dnamics excercises solutionPooja M
This document contains physics exercises related to rotational dynamics. It includes multiple choice questions testing concepts like angular velocity, moment of inertia, and rolling motion. It also includes longer questions requiring derivations of expressions for acceleration, speed, and minimum speeds for circular motion situations involving banked roads, conical pendulums, and rotational kinetic energy. The questions cover topics like conservation of angular momentum, radius of gyration, parallel axis theorem, and circular motion scenarios involving coins on spinning discs, ants on bicycle wheels, and cyclists in cylindrical wells.
DEWATERING-AN EFFECTIVE TOOL FOR SLOPE STABILIZATIONRathin Biswas
Stabilizing a slope is very critical issue for geotechnical engineering. Two of the main four Slope stabilization parameters viz. Rock mass Characteristics and stress on slope arc the inherent parameters for a particular slope. By a limit equilibrium analysis it can be observed that shallower slope cost a lot, thus De-watering is the best way for slope stabi Iization.
3.[15 25]modeling of flexural waves in a homogeneous isotropic rotating cylin...Alexander Decker
The document summarizes research on modeling flexural wave propagation in a homogeneous isotropic rotating cylindrical panel. Three displacement potential functions are introduced to uncouple the governing equations of motion. Boundary conditions are used to obtain frequency equations, which are then studied numerically for the material copper. Relative frequency shifts are plotted in the form of dispersion curves using MATLAB. The research aims to better understand wave propagation under rotation, which has applications in fields like civil, mechanical and marine engineering.
11.modeling of flexural waves in a homogeneous isotropic rotating cylindrical...Alexander Decker
The document summarizes research on modeling flexural wave propagation in a homogeneous isotropic rotating cylindrical panel. Three displacement potential functions are introduced to uncouple the governing equations of motion. Boundary conditions are used to obtain frequency equations, which are then studied numerically for the material copper. Relative frequency shifts are plotted in the form of dispersion curves using MATLAB. The research aims to better understand wave propagation under rotation, which has applications in fields like civil, mechanical and marine engineering.
This document discusses bending stresses in beams. It begins by defining key terms like deflection curve, plane of bending, and curvature. It then discusses the types of bending (pure and non-uniform), and how longitudinal strains in beams vary linearly with the distance from the neutral axis. An example calculation is provided to determine radius of curvature, curvature, and midpoint deflection given longitudinal strain. The document concludes by stating that for linear elastic materials, normal stress varies linearly with distance from the neutral axis, and two equilibrium equations must be satisfied since only a bending moment acts on the cross section.
The document discusses concepts related to shear force and bending moment in beams, including:
- Definitions of bending, beams, planar bending, and types of beams including simple, cantilever, and overhanging beams.
- Calculation sketches simplify beams, loads, and supports for analysis.
- Internal forces in bending include shear force and bending moment. Relations and diagrams relate these to external loads.
- Equations define shear force and bending moment at each beam section. Diagrams illustrate variations along the beam.
1) Lateral loads on a beam cause it to bend into a deflection curve. Pure bending occurs when the bending moment is constant, resulting in zero shear force. Non-uniform bending happens when the bending moment is variable, with non-zero shear force.
2) The radius of curvature and curvature of a beam are defined based on the deflection curve. Longitudinal strains in a beam vary linearly with distance from the neutral axis, causing elongation on one side and shortening on the other.
3) For a simply supported beam bent into a circular arc, the example calculates the radius of curvature, curvature, and midpoint deflection based on the given bottom strain. The normal stress is directly proportional to
This document gives the class notes of Unit-8: Torsion of circular shafts and elastic stability of columns. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Constraints restrict the motion of a system to fewer than three independent coordinates. Holonomic constraints can be expressed as mathematical equations involving the coordinates and time, while non-holonomic constraints cannot. Constraints introduce difficulties by making coordinates dependent and introducing constraint forces. For holonomic constraints, generalized coordinates can be used to eliminate dependent coordinates and reduce the number of degrees of freedom. D'Alembert's principle states that the sum of the work done by actual and constraint forces during virtual displacements is zero, and can be used to derive the Lagrangian equations of motion for constrained systems.
The aim of this paper is to compare the development of theoretical research on the collapse analysis of arches and vaults, with some significant constructions of arch bridges, in French and Italy during the XVIIIth and XIXth centuries. On this subject, the authors would develop a brief outline of most important researches about mechanical aspects of the arch bridge theory in the same centuries. Then it will be developed some considerations on the construction, behaviour and assessment of a little number of significant arch bridges, to verify the corresponding between construction, theoretical and mechanical approach, collapse mode and conservation approach of these architectures.
1) The document discusses shear force diagrams (SFD) and bending moment diagrams (BMD) for a loaded beam.
2) Key equations for a loaded beam include the sum of vertical forces equals zero, and the sum of clockwise and anticlockwise moments equals zero.
3) An example beam is given and the SFD and BMD are drawn by calculating reaction forces, shear forces, and bending moments at different points along the beam.
This document summarizes stress and strain concepts contributed by five authors from the Department of Geology at the University of Haripur. It defines stress and strain, related terminology, types of stress including normal and combined stresses, types of strain including tensile, compressive, shear and volumetric strains. It also describes Hooke's law which states that within the elastic limit, the ratio of stress to strain is constant, known as Young's modulus. Diagrams are included to illustrate different types of stresses and strains.
Baños es una ciudad ecuatoriana conocida por sus aguas termales y su proximidad al volcán Tungurahua. El documento recomienda varias actividades turísticas en la zona como visitar la Basílica de la Virgen de las Aguas Santas, las piscinas termales, cascadas cercanas e ir de excursión a Puyo en bicicleta. Baños ofrece una variedad de atracciones naturales y culturales en un paisaje andino con clima templado.
This document contains a chart with 3 data series showing varying values over time. Series 1 starts at a value of 5 and decreases over time, Series 2 begins at 4 and also decreases, and Series 3 starts at 3 and its values remain constant over time. The chart compares changes in the 3 data series.
Under repeated impact composite domes subjected 6 J energy, changes locally with
increasing drop height. The action of the dynamic load generates reactions at the
support and bending moments at points on the surface of the composite. The peak loads
were noted to increase and stabilise about some mean value; and the 150mm diameter
shell was more damage tolerant compared to the 200 mm diameter one.
Nextar ChemPharma Solutions provides integrated contract drug development and manufacturing services. They have over 35 experienced employees working in state-of-the-art laboratories and clean rooms. Their services include custom synthesis, formulation development, analytical testing, and GMP manufacturing of clinical trial materials. Nextar has successfully completed over 650 projects for 150 customers, developing over 50 innovative formulations and manufacturing over 25 products for clinical trials.
This document introduces new associated curves called k-principal direction curves and kN slant helices for spatial curves. It defines k-principal direction curves as integral curves of the k-th principle normal vector of the curve. A curve is a kN slant helix if its k-principal direction curve has constant geodesic curvature. The document establishes properties of the Frenet frame and curvature formulas for k-principal direction curves. It explores using these new curves to characterize different types of spatial curves.
Nurek rockfill dam (300 m). Problem of seismic safety of dam (4 p.)Yury Lyapichev
Serious problem of dam seismic safety was tried to be solved by reinforced concrete belt-elements incorporated in the upper part of upstream zone & clay core. But due to large construction settlements of this zone & core this solution was useless & expensive
This document provides an introduction to static floating stability of offshore structures. It defines key terms like center of buoyancy, center of gravity, metacenter and righting moment. It describes the principles of vertical and rotational equilibrium for floating bodies. As an example, it then calculates the static stability of a rectangular pontoon undergoing a lifting operation, finding the maximum angle of heel before the structure reaches equilibrium.
This document summarizes a research paper that develops a mathematical model for analyzing the three-dimensional shape of a long twisted rod hanging under gravity, such as a pipeline being laid from a barge. The model uses the geometrically exact theory of linear elastic rods and formulates the problem as a boundary value problem that is solved using matched asymptotic expansions. The truncated analytical solution is compared to results from a numerical scheme and shows good agreement. The method is then applied to consider the near-catenary shape of a clamped pipeline during the laying process.
Simulation of segregated flow over the 2 d cylinder using star ccm+Burak Turhan
In this thesis numerical simulation for classical case of flow over a cylinder is accomplished for 2D models using commercial CFD code Star CCM+ with k-ϵ model approach. The results are validated by comparing the Drag coefficients to the previously published data. The simulation is carried out to for Reynolds number 3900 to investigate the turbulence modeling on separation from curved surfaces of two different sizes of a circular cylinder, a cylinder with triangular cross section and a rectangular cross section. Investigation of different turbulence models and Mesh convergence is carried out.
The investigation of the turbulence model of the circular cylinder is carried out by the drag coefficient obtained by four different turbulence models such as K-Epsilon Turbulence, K-Omega Turbulence, Reynolds Stress Turbulence and Spalart-Allmaras Turbulence. Drag coefficient found out by different turbulence model is compared with the experimental value of a previously published data. The Mesh Convergence have been carried out by implementing different base mesh size in a decreasing order and the convergence is obtained when the drag coefficient becomes constant
The document discusses slope mass rating (SMR) and its use in assessing the stability of rock slopes. SMR is calculated based on the basic rock mass rating (RMR) minus adjustment factors (F1, F2, F3) that account for discontinuity orientation plus an additional factor (F4) depending on excavation method. SMR values are used to classify slope stability into five classes, with recommended support measures depending on the class such as bolting, shotcrete, or retaining walls. The document also discusses factors that can affect slope stability and adaptations made to the SMR system for use in different regions.
Chapter 1 - rotational dnamics excercises solutionPooja M
This document contains physics exercises related to rotational dynamics. It includes multiple choice questions testing concepts like angular velocity, moment of inertia, and rolling motion. It also includes longer questions requiring derivations of expressions for acceleration, speed, and minimum speeds for circular motion situations involving banked roads, conical pendulums, and rotational kinetic energy. The questions cover topics like conservation of angular momentum, radius of gyration, parallel axis theorem, and circular motion scenarios involving coins on spinning discs, ants on bicycle wheels, and cyclists in cylindrical wells.
DEWATERING-AN EFFECTIVE TOOL FOR SLOPE STABILIZATIONRathin Biswas
Stabilizing a slope is very critical issue for geotechnical engineering. Two of the main four Slope stabilization parameters viz. Rock mass Characteristics and stress on slope arc the inherent parameters for a particular slope. By a limit equilibrium analysis it can be observed that shallower slope cost a lot, thus De-watering is the best way for slope stabi Iization.
3.[15 25]modeling of flexural waves in a homogeneous isotropic rotating cylin...Alexander Decker
The document summarizes research on modeling flexural wave propagation in a homogeneous isotropic rotating cylindrical panel. Three displacement potential functions are introduced to uncouple the governing equations of motion. Boundary conditions are used to obtain frequency equations, which are then studied numerically for the material copper. Relative frequency shifts are plotted in the form of dispersion curves using MATLAB. The research aims to better understand wave propagation under rotation, which has applications in fields like civil, mechanical and marine engineering.
11.modeling of flexural waves in a homogeneous isotropic rotating cylindrical...Alexander Decker
The document summarizes research on modeling flexural wave propagation in a homogeneous isotropic rotating cylindrical panel. Three displacement potential functions are introduced to uncouple the governing equations of motion. Boundary conditions are used to obtain frequency equations, which are then studied numerically for the material copper. Relative frequency shifts are plotted in the form of dispersion curves using MATLAB. The research aims to better understand wave propagation under rotation, which has applications in fields like civil, mechanical and marine engineering.
This document discusses bending stresses in beams. It begins by defining key terms like deflection curve, plane of bending, and curvature. It then discusses the types of bending (pure and non-uniform), and how longitudinal strains in beams vary linearly with the distance from the neutral axis. An example calculation is provided to determine radius of curvature, curvature, and midpoint deflection given longitudinal strain. The document concludes by stating that for linear elastic materials, normal stress varies linearly with distance from the neutral axis, and two equilibrium equations must be satisfied since only a bending moment acts on the cross section.
The document discusses concepts related to shear force and bending moment in beams, including:
- Definitions of bending, beams, planar bending, and types of beams including simple, cantilever, and overhanging beams.
- Calculation sketches simplify beams, loads, and supports for analysis.
- Internal forces in bending include shear force and bending moment. Relations and diagrams relate these to external loads.
- Equations define shear force and bending moment at each beam section. Diagrams illustrate variations along the beam.
1) Lateral loads on a beam cause it to bend into a deflection curve. Pure bending occurs when the bending moment is constant, resulting in zero shear force. Non-uniform bending happens when the bending moment is variable, with non-zero shear force.
2) The radius of curvature and curvature of a beam are defined based on the deflection curve. Longitudinal strains in a beam vary linearly with distance from the neutral axis, causing elongation on one side and shortening on the other.
3) For a simply supported beam bent into a circular arc, the example calculates the radius of curvature, curvature, and midpoint deflection based on the given bottom strain. The normal stress is directly proportional to
This document gives the class notes of Unit-8: Torsion of circular shafts and elastic stability of columns. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
Constraints restrict the motion of a system to fewer than three independent coordinates. Holonomic constraints can be expressed as mathematical equations involving the coordinates and time, while non-holonomic constraints cannot. Constraints introduce difficulties by making coordinates dependent and introducing constraint forces. For holonomic constraints, generalized coordinates can be used to eliminate dependent coordinates and reduce the number of degrees of freedom. D'Alembert's principle states that the sum of the work done by actual and constraint forces during virtual displacements is zero, and can be used to derive the Lagrangian equations of motion for constrained systems.
The aim of this paper is to compare the development of theoretical research on the collapse analysis of arches and vaults, with some significant constructions of arch bridges, in French and Italy during the XVIIIth and XIXth centuries. On this subject, the authors would develop a brief outline of most important researches about mechanical aspects of the arch bridge theory in the same centuries. Then it will be developed some considerations on the construction, behaviour and assessment of a little number of significant arch bridges, to verify the corresponding between construction, theoretical and mechanical approach, collapse mode and conservation approach of these architectures.
1) The document discusses shear force diagrams (SFD) and bending moment diagrams (BMD) for a loaded beam.
2) Key equations for a loaded beam include the sum of vertical forces equals zero, and the sum of clockwise and anticlockwise moments equals zero.
3) An example beam is given and the SFD and BMD are drawn by calculating reaction forces, shear forces, and bending moments at different points along the beam.
This document summarizes stress and strain concepts contributed by five authors from the Department of Geology at the University of Haripur. It defines stress and strain, related terminology, types of stress including normal and combined stresses, types of strain including tensile, compressive, shear and volumetric strains. It also describes Hooke's law which states that within the elastic limit, the ratio of stress to strain is constant, known as Young's modulus. Diagrams are included to illustrate different types of stresses and strains.
Baños es una ciudad ecuatoriana conocida por sus aguas termales y su proximidad al volcán Tungurahua. El documento recomienda varias actividades turísticas en la zona como visitar la Basílica de la Virgen de las Aguas Santas, las piscinas termales, cascadas cercanas e ir de excursión a Puyo en bicicleta. Baños ofrece una variedad de atracciones naturales y culturales en un paisaje andino con clima templado.
This document contains a chart with 3 data series showing varying values over time. Series 1 starts at a value of 5 and decreases over time, Series 2 begins at 4 and also decreases, and Series 3 starts at 3 and its values remain constant over time. The chart compares changes in the 3 data series.
Under repeated impact composite domes subjected 6 J energy, changes locally with
increasing drop height. The action of the dynamic load generates reactions at the
support and bending moments at points on the surface of the composite. The peak loads
were noted to increase and stabilise about some mean value; and the 150mm diameter
shell was more damage tolerant compared to the 200 mm diameter one.
Nextar ChemPharma Solutions provides integrated contract drug development and manufacturing services. They have over 35 experienced employees working in state-of-the-art laboratories and clean rooms. Their services include custom synthesis, formulation development, analytical testing, and GMP manufacturing of clinical trial materials. Nextar has successfully completed over 650 projects for 150 customers, developing over 50 innovative formulations and manufacturing over 25 products for clinical trials.
This document inculcates the diversified range of Value Added Services offered by Kanban Infosystem Pvt. Ltd in addition to the generic services rendered by them to the clients.
Abstract: Potential functions and Fourier series method in the cylindrical coordinate system are employed to solve
the problem of moving loads on the surface of a cylindrical bore in an infinite elastic medium. The steady-state
dynamic equations of medium are uncoupled into Helmholtz equations, via given potentials. It is used that because
of the superseismic nature of the problem, two mach cones are formed and opened toward the rear of the front in the
medium. The stresses and displacements are obtained by using integral equations with certain boundary conditions.
Finally, the dynamic stresses and displacements for step loads with axisymmetric and nonaxisymmetric cases are
obtained and discussed in details via a numerical example. Moreover, effects of Mach numbers and poisson's ratio
of medium on the values of stresses are discussed.
El documento describe los materiales utilizados para fabricar un lápiz mediante coextrusión de tres termoplásticos. Estos materiales son: 1) la mina, 2) una capa intermedia de protección compatible con la mina y el material de madera y con un punto de fusión igual o superior, y 3) el material de madera expandida. Los tres materiales contienen el mismo componente base como un copolímero poliestireno acrílico.
The document provides the address and contact information for a church located at No.5 Kodiak Crescent, Unit 7 in Toronto, Ontario, Canada with the postal code L6A 3V8. It lists the church's telephone numbers as 905-417-7463 or 416-857-2375 and its email address as choifin@rogers.com.
modeling nonlinear vibration of two link flexible manipulator by finite elem...hamidmohsenimonfared
The document discusses the results of a study on the effects of a new drug on memory and cognitive function in older adults. The double-blind study involved 100 participants aged 65-80 and found that those given the drug performed significantly better on memory and problem-solving tests than the placebo group after 6 months. However, longer term effects beyond 6 months are still unknown and require further research.
This paper presents an algorithm for shape optimization of composite pressure
vessels head. The shape factor which is defined as the ratio of internal volume to weight of
the vessel is used as an objective function. Design constrains consist of the geometrical
limitations, winding conditions, and Tsai-Wu failure criterion. The geometry of dome shape
is defined by a B-spline rational curve. By altering the weights of control points, depth of
dome, and winding angle, the dome shape is changed. The proposed algorithm uses genetic
algorithm and finite element analysis to optimize the design parameters. The algorithm is
applied on a CNG pressure vessel and the results show that the proposed algorithm can
efficiently define the optimal dome shape. This algorithm is general and can be used for
general shape optimization
This document contains a chart with 3 data series showing varying values over time. Series 1 starts at a value of 5 and decreases over time, Series 2 starts at 4 and also decreases, and Series 3 fluctuates between values of 2 and 3 over the measured period.
The document discusses elasticity and seismic waves. It introduces key concepts such as strain, stress, deformation, Hooke's law, and the relationship between stress and strain. It describes how elasticity theory can be used to understand the propagation of elastic waves from a source, such as an earthquake, and how this relies on describing the kinematic deformation and resulting stresses using elastic constants. It also provides mathematical background on vectors, tensors, and their application to describing deformation.
Boundary Layer Flow in the Vicinity of the Forward Stagnation Point of the Sp...iosrjce
Exact solutions are important not only in its own right as solution of particular flows, but also serve as accuracy check for numerical solution. Exact solution of the Navier-Strokes equation are, for example, those
of steady and unsteady flows near a stagnation point, Stagnation point flows can either be viscous or inviscid,
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temperature and velocity fields for ξ = 0 are numerically computed. This shows that the thermal boundary layer
thickness decreases as Prandtl number Princreases.The surface heat transfer (28) increases with the Prandtl
number Pr. The surface heat transfer (28) at the starting of motion is found to be strangely dependent on the
Prandtl number Pr. But it is dependent of magnetic field, buoyancy force Bp and Rotation Parameter Ro.
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Physical Society the sentence but short speech on challenges of educational system in Bangladesh studies and sadie bahamians Horse hijabs husks jms bans bbs bbs bdb dnc dms msn hdhdhdbhdhdhdhdhdhdhdhdhdhfhdhdhdhdbdbdhdhdbdhdhdhdh
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GraphRAG for Life Science to increase LLM accuracy
121 9168life0902 812_822[1]
1. Life Science Journal 2012;9(2) http://www.lifesciencesite.com
Dynamic Stresses and Displacements around Cylindrical Cavities in an Infinite Elastic Medium under
Moving Step Loads on the Cavity's Surface
Hamid Mohsenimonfared 1 M.Nikkhahbahrami 2
1.
Department of mechanical Engineering , Science & Research Branch , Islamic Azad University(IAU), Tehran, Iran
2.
Department of mechanical Engineering ,Tehran University , Tehran , Iran
h-mohsenimonfared@ iau-arak.ac.ir
Abstract: Potential functions and Fourier series method in the cylindrical coordinate system are employed to solve
the problem of moving loads on the surface of a cylindrical bore in an infinite elastic medium. The steady-state
dynamic equations of medium are uncoupled into Helmholtz equations, via given potentials. It is used that because
of the superseismic nature of the problem, two mach cones are formed and opened toward the rear of the front in the
medium. The stresses and displacements are obtained by using integral equations with certain boundary conditions.
Finally, the dynamic stresses and displacements for step loads with axisymmetric and nonaxisymmetric cases are
obtained and discussed in details via a numerical example. Moreover, effects of Mach numbers and poisson's ratio
of medium on the values of stresses are discussed.
[Hamid Mohsenimonfared, M. Nikkhahbahrami. Dynamic Stresses and Displacements around Cylindrical
Cavities in an Infinite Elastic Medium under Moving Step Loads on the Cavity's Surface. Life Sci J
2012;9(2):812-822] (ISSN:1097-8135). http://www.lifesciencesite.com. 121
Keywords: Wave propagation, Fourier series method, cylindrical cavities, Helmholtz equations, moving load.
1. Introduction terms of Hankel functions, and finally the stresses
1.1 General Remarks and displacements are found by inversion of the
Moving loads on the surfaces have been transformed quantities. In this paper the coefficients
investigated by many researchers. Investigation on of the stresses and displacements are found by
dynamic stresses in solids is very significant in the solving sets of coupled integral equations.
study of dynamic strength of materials and in the The waves are expanded into Fourier series in
design of underground structures subject to ground terms of the angle, , around the opening. The
blasting waves. A related but considerably simpler stress field of the wave is written in terms of potential
problem has been treated by Biot (1952), who functions which satisfy the equations of motion.
considered space- harmonic axisymmetric standing These equations decoupled via introducing the
waves and obtained a closed form solution. Another potential functions and reduced to Helmholtz
related problem was treated by Cole and Huth (1958), equations that the potentials satisfy.
who considered a line load progressing with a These potential functions are in integral form
velocity V on the surface of an elastic half- space. with unknown functions in the integrands. Therefore
Because of the simpler geometry, they were able to the Fourier series coefficients of the stresses and
obtain a solution in closed form. Adrianus (2002) displacements are also in integral form with unknown
investigated the moving point load problem in soil integrands. The applied boundary tractions (the step
dynamics with a view to determine the ground loads) are expanded into a Fourier series in and
motion generated by a high-speed train traveling on expressions for the stress and displacement
a poorly consolidated soil with low shear wave speed. components at points in the medium are derived for
M.C.M. Bakker (1999) revisited the nonaxisymmet- each term of the Fourier series as functions of the
rical boundary value problem of a point load of radial distance r from the cavity axis and the distance
normal traction traveling over an elastic half-space. z behind the wave front.
M.Rahman (2001) considered the problem of a line The following three cases of step loads are
load moving at a constant transonic speed across the considered: normal to the surface, tangential to the
surface of an elastic half-space and derived solution surface in the direction of the axis of the bore, and
of the problem by using the method of Fourier tangential to the circle of load application. These
transform. Iavorskaia (1964) also studied diffraction results can be used, by superposition, to determine
of a plane longitudinal wave on circular cylinder. the effects of other load patterns moving with the
One basic method has been used for the solution of velocity V in the direction of the axis of the bore.
these problems, the solution is obtained by using an Numerical solution of these equations gives the
integral transform of the displacement potentials. The values of the unknown functions. These values can
resulting transformed equations are then solved in
812
2. Life Science Journal 2012;9(2) http://www.lifesciencesite.com
then be used to find the stresses and displacements on 2. Governing equations and general solutions
the boundary and also anywhere in the medium. Consider a cylindrical cavity of radius r = a in a
linearly elastic, homogeneous, and isotropic medium
1.2 Problem Description referred to a fixed coordinate system r, θ , z whose
The object of this work is to obtain stresses and origin lies on the axis of the cavity.
displacements in an elastic medium in the vicinity of A step load along the circle at z = -vt progresses
a cylindrical cavity which is engulfed by a plane along the interior of the cavity with a velocity V such
stress wave of dilatational travelling parallel to the that the stresses on the boundary r=a are:
axis of the cylinder, as shown in Figure 1.
The step load has an arbitrary distribution P (θ)
along the circumference of the circle and moves with
σrr σ (θ)U(z vt)
ra 1
(1)
a velocity V C1 C 2 ; therefore, the speed is σrθ
ra
σ 2 (θ ) U(z vt) (2)
σrz
superseismic with respect to both the dilatational and
shear waves in the medium. Consequently, the σ3 (θ ) U(z vt) (3)
disturbances which were initiated far behind the front ra
on the boundary of the cavity cannot reach the Where the functions σ k (θθ) define the
vicinity of the wave front for some time after the distribution of the applied load. To determine the
incident wave passes. steady state solution, a moving coordinate system (r ,
θ ,z) is introduced such that:
r r , θ θ , z z Vt (4)
The following treatment is restricted to the case
where the velocity V is greater than C1 and C2 , the
respective propagation velocities of dilatational and
equivoluminal waves in the medium. Hence
V V (5)
M 1 1 , M 1 2
C1 C2
Figure 1. Moving step load Where
λ 2μ μ (6)
Moreover, because of the super seismic nature of C1 , C2
ρ ρ
the problem, it should be expected that two mach
cones will be formed in the medium, as shown in The equations of motion in cylindrical coordinates, r ,
Figure 2. These cones should open toward the rear of , z, for an elastic medium, may be expressed in the
the front. Furthermore, there can be no stresses or following form:
displacements ahead of the leading front. u θ Δ ρ u r
2
1 λ
If a coordinate system is assumed to move along 2u r (u r 2 ) ( 1)
the cylinder with the wave front, it is seen that the r 2 θ μ r μ t 2
state of stress at points close behind the wave front
1 Δ ρ u θ
2
1 u r λ
depends only on relative position of them with 2uθ (u 2
2 θ
) ( 1)
respect to the front. Thus, in the vicinity of the wave r θ μ r θ μ t 2
Δ ρ u z
front, provided that the end of the cavity is far away, 2
λ
the problem may be treated as a steady-state case. In 2u z ( 1)
μ z μ t 2
other words, in the moving coordinate system, the
state of stress and displacement is independent of (7)
time.
Where the dilatation , Δ , and the laplacian operator,
2 , are given by:
u u 1 u θ u z
Δ r r
r r r θ z (8)
2 1 1 2 2
2
2
r r r r 2 θ 2 z 2
As mentioned earlier, the assumption of the existence
of a steady-state case and trans-formation form r ,
, z coordinates to r , , z results in elimination of
Figure 2. Geometry of the problem and the coordinate the time variable, t , from the equations of motion.
systems
813
3. Life Science Journal 2012;9(2) http://www.lifesciencesite.com
This transformation is performed by the following φ n 2ψn
n
relations, as given in relations (4): u n, r
z z Vt r rz r n
n n ψ n n
u n, θ φ n (13)
, V r r z r
z z t z φ n 2 1 n 2
Therefore equations (7) may be expressed as follows: u n, z ψ
z r 2 r r r 2 n
1 u θ λ Δ ρ 2 u r
2
2u r (u r 2 ) ( 1) V These equations may be obtained from the vector
2
r θ μ r μ z 2
equation
1 Δ ρ 2 u θ
2
1 u r λ
2uθ (uθ 2 ) ( 1) V u grad φ curl
r2 θ μ r θ μ z 2 Where is the sum of two independent vectors as
Δ ρ 2 u z
2
λ follows:
2u z ( 1) V
μ z μ z 2
curl ψ
(9)
Stress components are given by
The vectors and ψ have only one non- zero
u
σrr λΔ 2μ r component which is in the z- direction in both cases.
r
u 1 uθ r 0 ψr 0
σθθ λΔ 2μ ( r ) θ 0 ψθ 0
r r θ
z ψz ψ
u
σzz λΔ 2μ z (10)
z By substitution of the values given in equations (13)
1 u r u θ u θ into the equations (9), it can be shown that the
σ rθ μ( )
r θ r r potential functions satisfy the modified wave
u u equations.
σ rz μ( r z )
z r V 2 φn
2
u 1 u z 2φ n ( )
σθz μ( θ ) c1 z 2
z r θ
V ψn
2
(14)
Displacement components u r , u θ and u z may be 2ψn ( )2
expressed in Fourier series:
c2 z 2
V 2 χn
2
χn ( )
2
u r (r,θ, z) u n, r (r, z)cosnθ c2 z 2
n0 Stress components are expressed in Fourier series
(11)
u θ (r,θ, z) u n, θ (r, z)sinnθ form as follows:
n 1
σ rr (r,θ, z) σ n,rr (r, z) cos nθ
u z (r,θ, z) u n, z (r, z)cosnθ n 0
n0
Three potential functions are now introduced, σθθ (r,θ, z) σn,θθ (r, z) cos nθ
n 0
φ(r, θ, z) φ n (r, z)cosnθ
n 0
σzz (r,θ, z) σn,zz (r, z) cos nθ
n 0
ψ(r, θ, z) ψ n (r, z)cosnθ (12) (15)
n 0 σrθ (r,θ, z) σn,rθ (r, z) sinnθ
n 1
(r,θ, z) n (r, z)sinnθ
n 1
σ rz (r,θ, z) σ n,rz (r, z) cos nθ
The displacement components u n, r , u n, θ and u n, z n 0
are defined as follows: σθz (r,θ, z) σ n,θz (r, z) sin nθ
n 1
Equations (11) and (15) may be substituted into
equations (10), and as a result stress- displacement
relations may be written for each term of the series:
u n,r (16)
σ n,rr λΔ 2 μ
r
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u n,r u n,θ β1 M1 1
2 2
, β2 M2 1
σ n,θθ λΔ 2 μ
r n r
2 2
The differential equations (14) may be written in the
following form:
u n,z
σ n,zz λΔ 2 μ 2φn 1 φn n2 2φn
z 2 φn β12
u n,r u θ u n,θ R2 R R R Z2
σ n,rθ μ n
r r r 2ψn 1 ψn n2 2ψn
2 ψn β22 (20)
u n,r u n,z R2 R R R Z2
σ n,rz μ
z r 2 n 1 n n 2 2 n
2
2 n β2
R 2 R R R Z 2
u n,θ u n,z
σ n,θz μ
z n r
It is seen that these equations have the same general
form as the differential equations of the cylindrical
where waves obtained in reference (13). Therefore solutions
of equations (20) may be obtained in a manner
u n,r u n,r u n,θ u n,z similar to that in reference (13). These solutions are
Δ n given in integral form as follows (see Appendix A for
r r r z
verification of the solutions):
Substitution of equations (13) into equations (16) and
∞
application of the differential equations (14) result in φn ∫ fn (Z - Rβ1coshu)coshnu du
0
the following equations for stress components:
(21)
ψn 0 gn (Z Rβ2coshu)coshnudu
a 2 σ n,rr
M 2 2M1
φ n,ZZ 2 φ n,RR 2ψ n,RRZ
2
2
μ n 0 h n (Z Rβ 2coshu)coshnu du
2n 1
n,R n From consideration of the fact that the
R R
disturbances are zero ahead of the wave front, it is
a 2 σ n,θθ
μ
M 2 2 φ n,ZZ 2 φ n,RR
2
seen that the functions fn , gn and hn are zero for the
values of their arguments less than
2
2 Rβ1 , Rβ2 and Rβ2 ,respectively.
ψ n,RZ n ψ n,Z 2n n,R 1 n
Therefore the upper limits of the integrals may be
R
R R
R changed from to the following values:
a 2 σ n,zz
μ
M 2 2M1 2 φ n,ZZ 2 M 2 1 ψ n,ZZZ
2
2
2 u1 cosh-1(1
Z
Rβ1
) for φn
(22)
Z
a 2 σ n,rθ 2n
(17) -1
u2 cosh (1 ) for ψn & n
1 2n 1 Rβ2
φ n,R φ n ψ n,RZ ψ n,Z
μ R R
R R
The integrals are then written with these limits:
M 2 1 n,ZZ 2 n,RR
2
u
φn 01 fn ZRβ1coshu coshnudu
a 2 σ n,rz
μ
2φ n,RZ
M2 2
2 ψ n,RZZ
n
n,Z
R u2
ψn 0 g n ZRβ2 coshucoshnudu (23)
n 0 hn ZRβ2 coshucoshnudu
2
a σ n,θz 2n n u
φ n,Z M 2 2 ψ n,ZZ n,RZ
2
2
μ R R
Where the second set of subscripts of
3. Expressions of stresses and displacements
φn ,ψn and n represent the partial derivatives of
Substitution of equations (23) into equations (17)
these functions. R and Z are the dimensionless gives the following expressions for stress
variables form: components.
r z (18)
R , Z
a a a2σn,rr
0 1 f (η1)[M22 2
u
The values M1 and M2 are defined as follows: μ
2β12sinh 2u]cosh nu du (24a)
V V
M1 , M2 2β22 0 2 g(η2 )cosh2u coshnu du
u
C1 C2 2 u2 h(η )sinh 2u coshnu du
(19) β2 0 2
Let
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a2σn,θθ stress components must satisfy the following
0 1 f (η1)[M22 2
u
boundary conditions:
μ
2β12cosh2u] cosh nu du (24b)
σ n,rr r a σ n1u(Z)
2β22 0 2 g(η2 )sinh2u cosh nu du
u σ n,rz r a σ n2 u(Z) (26)
β2 0 h(η2 )sinh 2ucosh nu du σ n,rθ r a σ n3 u(Z)
2 u2
These equations are satisfied for each term n. The
a 2σ n,zz coefficients of the stress, σ n,rr , σ n,rθ and σ n,rz are
(M 2 2
2M12 2)0 1 f (η1 )cosh nu du
u
(24c)
μ
expressed in equations (24) in integral form. These
2β 2 0 g(η2 )cosh nu du
2 u2
integrals include the unknown functions
a 2σ n,rθ f (η1 ), g (η 2 ) and h (η 2 ) which are to be found by
β12 0 1 f (η1 )sinh 2u sinh nu du
u
solving the set of three simultaneous integral
μ (24d)
β 2 2 0 2 g(η2 )sinh 2u sinh nu du
u equations. These values then may be substituted back
β 2 0 h(η2 )cosh 2u cosh nu du
2 u2 into the equations (24) and (25) to find the stress
components and displacement components of the
a2σ n,rz waves at any point on the boundary or in the medium
2β1 u1 f (η1)coshu cosh nu du
μ 0 (24e) behind the move front.
2 u 2
β2 (M2 2) 0 g (η2 )coshu coshnu du 5. Solution of the Boundary Equations
β2 0 2 h(η2 )sinh u sinh nu du
u Numerical solution of the boundary equations
requires finding numerical values of the functions
a2σn,θz f (η1 ), g (η 2 ) and h (η 2 ) .In the following para-
2β10 1 f (η1)sinh u sinh nu du
u
μ (24f) graph, the changes in variables are used. At the
β2 (M22 2) 0 2 g(η2 )sinh u sinh nu du
u
boundary, the radius R is fixed, R=1. Therefore the
β2 0 2 h(η2 )coshu coshnu du
u arguments of the functions f , g and h are:
Substitution of equations (23) into equations (13) η1 Z β1 coshu (27)
gives the following expressions for displacement η2 Z β 2 cos hu
components, (28)
Z Z
We let : ξ1 , ξ2 kξ1
β1 β2
μu n,r Rβ12 u1 f (η1 ) sinh u [n sinh nu cosh u
0 sinh u cosh nu] du
a n2 1
Rβ 2 2 u 2 g(η2 ) sinh u [n sinh nu cosh u Where K β1 /β 2 . Equations (27) may be written
0 sinh u cosh nu] du (25a)
n2 1 as:
2
Rβ 2 u 2 h(η2 ) sinh u [sinh nu cosh u
η1 β1 ξ1 coshu
n2 1
0 n sinh u cosh nu] du
η2 β 2 kξ1 cos hu
μu n,θ Rβ12 u f (η1) sinh u [n sinh u cosh nu
A new variable is now introduced by the following
0 1 sinhn u cosh u] du
2
a n 1 relations:
Rβ22 u g(η2 ) sinh u [n sinh u cosh nu (25b)
0 2 sinh nu cosh u] du
n2 1 coshu 1 ξ
Rβ22 u 2 h(η2 ) sinh u[n sinh nu cosh u dξ dξ
0 du (29)
n2 1 sinh u cosh nu] du sin hu 2ξ ξ 2
μu n,z Rβ1 u 1
f (η1) sinhuSinhn u du
a n 0 (25c) The limits of the integrals with this variable are as
- Rβ 23 u 2
0 g(η2 ) sinh usinh nu du follows:
n
Low er limits, u 0 , ξ 0
η1 Z Rβ1 cosh u u u1 , ξ ξ1 (30)
Where Upper limits
η2 Z Rβ 2 cosh u
u u2 , ξ ξ 2 kξ1
And f(η1), g(η2 ),h(η2 ) fn (η1), gn (η2 ), hn (η2 ) and
The upper limits are linear functions of Z and ξ 1 ;
primes represent the derivatives of the functions with
therefore, in order to perform numerical integration,
respect to their arguments.
the longitudinal axis ξ 1 is divided into small steps.
4. Boundary conditions
In order to satisfy the condition of a traction At every point along this axis the numerical
boundary at the face of the cavity, r=a , three of the integration is performed and at each step only one
new value of the functions
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f (η1 ), g(η2 ) and h(η2 ) enters into the σ rθ r a σ n3 sin nθ U (Z)
computations.
K=3 σ rr r a σ rz r a 0
As an example of the procedure of the numerical The curves are shown for two sets of prameters:
integration, the component of the stress in the radial V
direction is given symbolically below: Case 1: M1 2 ; υ 0.25
C1
σn,rr IF I G IH
rr rr rr
(31)
V
Case 2: M1 1.033 ; υ 0.25
Where
C1
IF u1 f η1 M2 2 2 β1 sinh 2u cos hnudu
rr 2
2
The values of M1 were chosen for application of
0
Irr 2β2 u2 gη2 cos h2ucos hnudu
G (32) the results to problems of some practical interest. The
2 0 stress components in each case approach the static
Irr β2 h η2 sin h2u sinhnududu
H 2 u2
plain strain solution as Z approaches infinity,
0
indicating that mathematical model produces correct
The integrals IF ,Irr and IH at the pth step are
rr
G
rr results for propagation of waves in the isotropic
expressed by using the convolution theorem in medium. For those cases in which the static solutions
summation form as follows: do not vanish, a typical overshoot above the value of
rr f R f R rr f
IF R F
rr 0 p
P 1
m 1
F
rr m Pm
F
p 0
the static (long term) solutions is observed.
Moreover, a decrease in the Mach number M1
R g R g R rr g
P 1 (33) appears to compress the stress response curve into a
G G
IG
rr rr 0 p
G
rr m Pm 0
p smaller range of Z such that the asymptotic values of
m 1
R h R h R rr h
P 1
H H the lower value M1=1,033 are obtained for smaller
IH
rr rr 0 p
H
rr m Pm 0
p
m 1 values of Z. Figures 12 and 13 show the stress
Where (f) , (g) and (h) are the unknown functions to components σ 0, θθ and σ 0, zz at the cavity boundary
be evaluated.
At this stage of integration the values (f)m , (g)m , r=a for the axisymmetric loading case, n=0, for the
(h)m are known for m=0 to p-1.The only unknowns in mach numbers M1=1.033, 1.5 and 2. As in the cases
these expressions are (f)p , (g)p , and (h)p . Similar where n 0, the stress components in each case
expressions are written for the other components of approach the static plain solutions as Z approaches
stress, at the pth step. The boundary conditions are infinity. Figures 14 through 19 show the
now in the form of a set of three simultaneous linear displacement components u n,r , u n,θ and u n,Z (n=1,
equations. Solution of this set results in the values of 2, 3, 4) for each of the three loading cases, k=1, 2, 3.
(f)p , (g)p , and (h)p . The procedure is then carried on Figures 20 through 22 show the u 0,r and u 0, Z
to the (P+1)th step; Similar operations are performed displacement components for the case n=0. These
to find the values of (f)p+1 , (g)p+1 , and (h)p+1 . displacement results are shown for the M1= 2,
As mentioned previously, when the values f, g and h 1 / 4 case only.
are found at each step, these values are substituted The only property of the material in the medium
into the expressions for ,u ,u and u which enters into computations is its poisson's ratio.
n,r n,θ n, z
σ n, θθ , σ n, zz , σ n, θz to compute the numerical Figures 23 through 26 represent the effect of this
parameter on the values of stress components for the
values of these stresses and displacements in the axisymmetric loading case, n=0. The following
medium. values of poisson's ratio are used in this study:
6. Numerical Results and Conclusion υ 0 , 0.15, 0.25 and 0.35
For the non-axisymmetric loadings characterized It is noticed that the change in poisson's ratio
by n > 0, numerical values of the stress components does not have a large effect on the maximum value of
σ n,θθ , σ n,zz and σ n,zθ at the cavity boundary r=a longitudinal stress for the load case k=2 (Figure 26),
are presented in this section. These stresses are given while it affects considerably the value of longitudinal
for the cases n=1, 2 for each of the three step- traction stress for the case load, k=1 (Figure 25) , and hoop
loading indicated below: stress for the two cases, k=1,2 (Figures 23 and 24) for
Index Applied load smaller values of Z.
σ rr r a σ n1 cos nθ U (Z)
K=1 σ rz r a σ rθ r a 0
σ rz r a σ n2 cos nθ U (Z)
K=2 σ rr r a σ rθ r a 0
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σ n,
0.8
1.
4
1.2
σ n1
1 0.6
Static Value
0.8
Static Value
0.6 n=1(M1=2) 0.4
n=1(M1=1.033)
0.4 n=2(M1=2) 0.25
n=2(M1=1.033)
0.2
Static Value
0.2
0 Z=z/a
0 4 8 12 16 20 24 28
-0.2
Static Value
0
Z=z/a
-0.4
0 4 8 1 1 2 2 2
-0.6 2 6
n=1(M1=2)0 4 8
-0.2 n=2(M1=2)
σ n,zz 0.25 n=1(M1=1.033)
n=2(M1=1.033)
σ n1
Figure 3. Stress σ at boundary due to step load ;
-0.4
σ rr σ n1 cos n u ( Z ); n 1,2
0 4 8 12 16
Figure 6. Stress σ zz at boundary due to step load ;
0.8
σ rr σ n1 cos n u ( Z ); n 1,2
0.4
Z=z/a
0 4 8 12 16 20 24 28
Static Value 0.5
0 Z=z/a
0
Static Value
n=1(M1=1.033) -0.5
n=1(M1=2) 0.25
-0.4 -1
0.25 n=2(M1=1.033)
-1.5 n=1(M1=2)
n=2(M1=2)
n=2(M1=2)
-2 n=1(M1=1.033)
-0.8
σ n, -2.5
n=2(M1=1.033)
σ n2 -3
-1.2 -3.5 σ n, zz
-4
σ n2
Figure 4. Stress σ at boundary due to step load : -4.5
σ rz σ n2 cos n u ( Z ); n 1,2
Figure 7. Stress σ zz at boundary due to step load ;
0 4 8 12 16 20 24 28 32 36 40 44 48 Z=z/a
0 σ rz σ n2 cos n u ( Z ); n 1,2
-0.5
σ n, zz 0.2
Static Value σ n3 0.1
-1 0 Z=z/a
-0.1
n=1(M1=2) Static Value
0.25 n=2(M1=2)
n=1(M1=1.033)
-0.2
-1.5
n=2(M1=1.033) -0.3
0.25
-0.4
Static Value
Static Value -0.5
-2
-0.6
n =1(M1=2)
σ n,θθ -0.7 n =2(M1=2)
n =1(M1=1.033)
-σ -0.8
n3 n =2(M1=1.033)
2.5
-0.9
0 4 8 12 16 20 24 28
Figure 5. Stress σ at boundary due to step load ;
σ r σ n3 sin n u ( Z ); n 1,2
Figure 8. Stress σ zz at boundary due to step load ;
σ r σ n3 sin n u ( Z ); n 1,2
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σ n,z ij , 0
0.4 1
σ n1 01
0.8
0.35
0.6
0.3
0.4
0.25 0.2
Static Value
Z=z/a
0.2 0 0
4 8 1 16 20
n=1(M1=2) -0.2 2M1=1.033
0.15
n=2(M1=2) M1=1.033
n=1(M1=1.033) -0.4 M1=1.5
0.1 n=2(M1=1.033) M1=1.5
-0.6 M1=2
0.05 M1=2
Static Value -0.8 Static Value
0 -1
0 4 8 12 16 20 24 28 32 36 40 44 48
-0.05 -1.2
-1.4
-0.1
0.25 Figure 12. Stresses at boundary due to
Figure 9. Stress σ z at boundary due to step load ; axisymmetric step load
σ rr σ n1u ( Z); n 0
σ rr σ n1 cos n u ( Z ); n 1,2
1.4 ij , 0
02
1.2 1
Static 0.5
1 Value 0 Z=z/a
0 4 8 16 20
n=1(M1=2) -0.5 12
0.8
n=2(M1=2) -1
M1=1.033
n=1(M1=1.033)
M1=1.5
-1.5
0.6
n=2(M1=1.033) 0.25 M1=2
-2 M1=1.033
0.4 M1=1.5
-2.5
M1=2
-3
0.2
-3.5
0 -4
0 4 8 12 16 20 24 28 -4.5
0.25 Z=z/a
Figure 10. Stress σ z at boundary due to step Figure 13. Stresses at boundary due to
axisymmetric step load
load ; σ rz σ n2 cos n u(Z ); n 1,2
σrz σn2 u ( Z ); n 0
0.1
4 8 12 16 20 24 Z=z/a
2
8
Static Value 0.00 0
0 Z=z/a
0 4 8 12 16 20 24 28 -0.10
n =4
-0.1 -0.20
-0.2 -0.30
n =3
-0.3
-0.40
STATIC
-0.50 VALUE
-0.4 n =2
n=1(M1=2) -0.60
-0.5 n=2(M1=2)
-0.70
n=1(M1=1.03
-0.6 3)
n=2(M1=1.033) -0.80 n =1
-0.90
-0.7 -1.00
-0.8 -1.10
σ n,z
un, r Static Value = -1.45
σ n3 0.25
a 01
Figure 11. Stress σz at boundary due to step load ;
Figure 14. Boundary displacement u n, r due to step
σ r σ n3 sin n u( Z ); n 1,2
load σ rr σ n1 cos n u ( Z ); n 1,2,3,4 , M 2 , 0.25 ,
1
r rz 0
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un, r un, z 0 4 8 12 16 20 24 28
a 02 a 01 0 Z=z/a
0.05 -0.1
-0.2
Static Value
0.04
-0.3
-0.4
0.03 n=1
-0.5
n=2 Static Value
0.02 n=3 -0.6
n=4 -0.7
0.01 -0.8
n=1
Static Value -0.9 n=2
0 Z=z/a -1 n=3
n=4
-1.1
-0.01
-1.2
-0.02 un,
0 4 8 12 16 20 24 28 a
03
Figure 15. Boundary displacements due to step load Figure 18. Boundary displacements due to step
;n=1,2,3,4 , M 2 , 0.25
1
load ;n=1,2,3,4 , M1 2 , 0.25
u n , r u n ,
a 03 a 01
Static Value = 1.45
0 4 8 12 16 20 24 28
0.9 Z=z/a
0
0.8 n=1 -0.1
n=2
0.7
n=3 -0.2
n=4 -0.3
0.6 -0.4 Static Value
-0.5
0.5 -0.6
n=1
-0.7
0.4 Static Value n=2
-0.8 n=3
-0.9 n=4
0.3
Static Value
-1
0.2
Static Value -1.1
0.1 -1.2
-1.3
Z=z/a
0 u n, z
0 4 8 12 16 20 24 28 a 02
Figure 19. Boundary displacements due to step
Figure 16. Boundary displacements due to step load
load ;n=1,2,3,4 , M1 2 , 0.25
;n=1,2,3,4 , M1 2 , 0.25
u
0 , r μ
u n, u n, z a σ
01
a 02 a 03 -0.7
0.028
-0.6
0.024
-0.5 STATIC
0.02
VALUE
0.016 n=1 -0.4
n=2
0.012 n=3
n=4
-0.3
0.008
-0.2
0.004
Static Value
-0.1
0
Z=z/a
0 4 8 12 16 20 24 28
-0.004 0
0 4 8 12 16 20 24 28
-0.008 Z=z/a
-0.012
Figure 20. Boundary displacement u due to step
-0.016 0, r
load σ01 ; n=0 , M1 2 , 0.25
Figure 17. Boundary displacements due to step
load ;n=1,2,3,4 , M 1 2 , 0.25
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u0, r u0, z
0 ,
a 02 a 01
02
0.08
0 . 25 , M 2
U ( Z )
0.3
rr 01
on r a
0.07 0
rz 0.2
0
rr on r a
0.06
U ( Z )
rz 02 0.1
0.05
0 z/a
0 4 8 12 16 20 24 28
0.04 -0.1
0
0.03 -0.2
0.02 -0.3 0.15
0.01 -0.4
0.00
0 4 8 12 16 20 24 28 32 36 40
Z=z/a Figure 24. Comparision of Hoop stress for different values of
poisson's ratio at boundary r=a due to axisymmetric step load
Figure 21. Boundary displacement u due to step σ02 ; n= 0,M1=2
0, r
0 , zz
load σ02 ; n= 0 u due to step load σ01 ; n= 0 01
0, z
, M 2 , 0.25
0.6
1
0.5
u 0,z μ 0.4
a σ 02 0
0.3 0.15
-3.2 0.25
0.2 0.35
-2.4 0.1
z/a
0
-1.6 0 4 8 12 16 20 24 28
-0.1
-0.8 -0.2
0 Figure 25. Comparision of Longitudinal stress for different
4 8 12 16 20 24 28 32 36 40 values of poisson's ratio at boundary r=a due to axisymmetric
Z=z/a
step load σ 01 ; n= 0, M1=2
Figure 22. Boundary displacement u due to step 0 , zz
0, z
load σ 02 ; n= 0 02
0 4 8 12 16 20 24 28
0 ,
0
01 z/a
0.8 -0.1
0.6 0 -0.2
0.4 0.15
0.2 0.25 -0.3
0.35 z/a
0 -0.4 0
-0.2 0 4 8 12 16 20 24 28 -0.5
-0.4 0.15
-0.6 -0.6
-0.8 -0.7
0.25
-1
-0.8 0.35
-1.2
-1.4 -0.9
-1.6 -1
Figure 23. Comparision of Hoop stress for different values of Figure 26. Comparision of Longitudinal stress for different
poisson's ratio at boundary r=a due to axisymmetric step load values of poisson's ratio at boundary r=a due to axisymmetric
σ01 ; n= 0,M1=2 step load σ 02 ; n= 0, M1=2
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