1. Thick cylinders are defined as cylinders where the ratio of thickness to internal diameter is greater than 1/20. For thick cylinders, the hoop stress varies across the thickness from a maximum at the inner circumference to a minimum at the outer circumference.
2. Equations are derived for the radial pressure (px) and hoop stress (σx) at any point in a thick cylindrical shell subjected to internal fluid pressure. These equations, known as Lame's equations, define px and σx as functions of radius x.
3. An alternate method for finding stresses in a thick cylinder is presented, where circumferential, longitudinal, and radial strains are related to stresses and Poisson's ratio.
This document is a technical report by Bibin Chidambaranathan on bending and shear stress distributions in mechanical engineering structures. It contains over 100 pages analyzing stress distributions in various standard structural sections including rectangular, circular, T, I, and composite sections. Equations for calculating bending stress and shear stress at different points within each cross section are presented along with diagrams illustrating the stress distributions.
This document discusses the elongation of a uniformly tapering circular rod under an axial tensile load. It defines the geometry of the rod, including the larger diameter D1, smaller diameter D2, and total length L. It then derives equations for the diameter Dx, cross-sectional area Ax, stress σx, strain εx, incremental change in length δlx, and total elongation δl of the rod. As an example, it calculates the modulus of elasticity E of a rod with given D1, D2, L, load P, and elongation. The key results are that the diameter varies linearly with distance x from D1, and the total elongation is given by δl = 4PL/(πE
The document derives expressions for Young's modulus in terms of other elastic constants like modulus of rigidity and bulk modulus. It considers a solid cube subjected to shear forces, which causes diagonal strains. Using relationships between shear strain, normal stress, longitudinal strain, and lateral strain, it arrives at the expression E = 2G(1+μ), where E is Young's modulus, G is modulus of rigidity, and μ is Poisson's ratio. It then considers a cube under mutually perpendicular tensile stresses and derives another expression for E in terms of bulk modulus.
The document discusses thermal stresses in a circular bar with a tapering circular cross-section that is fixed at both ends and subjected to an increase in temperature. It presents the following key points:
1) The bar will tend to expand due to the temperature increase but be prevented from doing so by being fixed at both ends, causing compressive stress.
2) An equation is derived relating the load required to counteract the expansion to the material properties, length, diameters at each end, and temperature change.
3) This equation is used to calculate that the maximum stress is directly proportional to the temperature change, Young's modulus, and larger diameter and inversely proportional to the smaller diameter.
4) As
Pressure research in kriss tilt effect 04122018 ver1.67Gigin Ginanjar
The document discusses tilt effects in high pressure pressure balances up to 500 MPa. It analyzes absolute and relative tilt effects through theoretical approaches, 2D and 3D FEA simulations, and experiments. The theoretical approach shows tilt can cause a change in effective area of around 3 ppm for 500 MPa balances and 1 ppm for 100 MPa balances. FEA simulations in perpendicular and tilted conditions were performed to investigate piston tilt effects on pressure distribution and effective area calculations. Experiments on various pressure balances showed 100 MPa balances follow a cosine behavior with tilt, while 500 MPa balances deviate more from ideal behavior.
CR-Submanifolds of a Nearly Hyperbolic Cosymplectic Manifold with Semi-Symmet...iosrjce
We consider a nearly hyperbolic cosymplectic manifold and we study some properties of CRsubmanifolds
of a nearly cosymplectic manifold with a semi-symmetric semi-metric connection. We also obtain
some results on 휉−horizontal and 휉 −vertical CR- submanifolds of a nearly cosymplectic manifold with a semisymmetric
semi-metric connection and study parallel distributions on nearly hyperbolic cosymplectic manifold
with a semi-symmetric semi-metric connection.
Cr-Submanifolds of a Nearly Hyperbolic Kenmotsu Manifold Admitting a Quater S...IJERA Editor
We consider a nearly hyperbolic Kenmotsu manifold with a quarter symmetric metric connection and study CRsubmanifolds
of a nearly hyperbolic Kenmotsu manifold with quarter symmetric metric connection. We also
study parallel distributions on nearly hyperbolic Kenmotsu manifold with quarter symmetric metric connection
and find the integrability conditions of some distributions on nearly hyperbolic Kenmotsu manifold with quarter
symmetric metric connection.
This document is a technical report by Bibin Chidambaranathan on bending and shear stress distributions in mechanical engineering structures. It contains over 100 pages analyzing stress distributions in various standard structural sections including rectangular, circular, T, I, and composite sections. Equations for calculating bending stress and shear stress at different points within each cross section are presented along with diagrams illustrating the stress distributions.
This document discusses the elongation of a uniformly tapering circular rod under an axial tensile load. It defines the geometry of the rod, including the larger diameter D1, smaller diameter D2, and total length L. It then derives equations for the diameter Dx, cross-sectional area Ax, stress σx, strain εx, incremental change in length δlx, and total elongation δl of the rod. As an example, it calculates the modulus of elasticity E of a rod with given D1, D2, L, load P, and elongation. The key results are that the diameter varies linearly with distance x from D1, and the total elongation is given by δl = 4PL/(πE
The document derives expressions for Young's modulus in terms of other elastic constants like modulus of rigidity and bulk modulus. It considers a solid cube subjected to shear forces, which causes diagonal strains. Using relationships between shear strain, normal stress, longitudinal strain, and lateral strain, it arrives at the expression E = 2G(1+μ), where E is Young's modulus, G is modulus of rigidity, and μ is Poisson's ratio. It then considers a cube under mutually perpendicular tensile stresses and derives another expression for E in terms of bulk modulus.
The document discusses thermal stresses in a circular bar with a tapering circular cross-section that is fixed at both ends and subjected to an increase in temperature. It presents the following key points:
1) The bar will tend to expand due to the temperature increase but be prevented from doing so by being fixed at both ends, causing compressive stress.
2) An equation is derived relating the load required to counteract the expansion to the material properties, length, diameters at each end, and temperature change.
3) This equation is used to calculate that the maximum stress is directly proportional to the temperature change, Young's modulus, and larger diameter and inversely proportional to the smaller diameter.
4) As
Pressure research in kriss tilt effect 04122018 ver1.67Gigin Ginanjar
The document discusses tilt effects in high pressure pressure balances up to 500 MPa. It analyzes absolute and relative tilt effects through theoretical approaches, 2D and 3D FEA simulations, and experiments. The theoretical approach shows tilt can cause a change in effective area of around 3 ppm for 500 MPa balances and 1 ppm for 100 MPa balances. FEA simulations in perpendicular and tilted conditions were performed to investigate piston tilt effects on pressure distribution and effective area calculations. Experiments on various pressure balances showed 100 MPa balances follow a cosine behavior with tilt, while 500 MPa balances deviate more from ideal behavior.
CR-Submanifolds of a Nearly Hyperbolic Cosymplectic Manifold with Semi-Symmet...iosrjce
We consider a nearly hyperbolic cosymplectic manifold and we study some properties of CRsubmanifolds
of a nearly cosymplectic manifold with a semi-symmetric semi-metric connection. We also obtain
some results on 휉−horizontal and 휉 −vertical CR- submanifolds of a nearly cosymplectic manifold with a semisymmetric
semi-metric connection and study parallel distributions on nearly hyperbolic cosymplectic manifold
with a semi-symmetric semi-metric connection.
Cr-Submanifolds of a Nearly Hyperbolic Kenmotsu Manifold Admitting a Quater S...IJERA Editor
We consider a nearly hyperbolic Kenmotsu manifold with a quarter symmetric metric connection and study CRsubmanifolds
of a nearly hyperbolic Kenmotsu manifold with quarter symmetric metric connection. We also
study parallel distributions on nearly hyperbolic Kenmotsu manifold with quarter symmetric metric connection
and find the integrability conditions of some distributions on nearly hyperbolic Kenmotsu manifold with quarter
symmetric metric connection.
This document presents two problems related to calculating elastic constants for mechanical structures. The first problem calculates the Poisson's ratio and modulus of elasticity for a bar where lateral strain is given. The second problem calculates the modulus of rigidity, bulk modulus, and change in volume for a cylindrical bar under hydrostatic pressure, given the longitudinal and lateral strains. Formulas for stress, strain, elastic constants, volume, and volumetric strain are provided. The values of the elastic constants and change in volume are calculated for each problem.
CR- Submanifoldsof a Nearly Hyperbolic Cosymplectic ManifoldIOSR Journals
This document discusses properties of CR-submanifolds of a nearly hyperbolic cosymplectic manifold. It begins with definitions of nearly hyperbolic cosymplectic manifolds and CR-submanifolds. The paper then obtains some results about the horizontal and vertical distributions on CR-submanifolds, including that the horizontal distribution is invariant by the structure tensor φ, while the vertical distribution is anti-invariant by φ. Lemmas are proved relating the covariant derivatives of φ on the horizontal and vertical distributions to terms involving the second fundamental form and normal connection.
This document presents the analysis of non-lifting potential flow past a thin symmetric hydrofoil using a finite difference method. The objectives are to solve the potential flow problem around a 2D hydrofoil and calculate the pressure distribution. A NACA 0012 hydrofoil is used. Governing equations and boundary conditions for the potential flow are described. A structured algebraic grid is used to generate points around the hydrofoil. Finite difference equations are derived and discretized on the grid points. The pressure coefficient is then calculated and compared to experimental results, showing good agreement.
Estimation Theory Class (Summary and Revision)Ahmad Gomaa
Summary of important theories and formulas in Estimation theory:
1) Cramer-Rao lower bound (CRLB)
2) Linear Model
3) Best Linear Unbiased Estimate (BLUE)
4) Maximum Likelihood Estimation (MLE)
5) Least Squares Estimation (LSE)
6) Bayesian Estimation and MMSE estimation
The document discusses the elongation of a bar hanging freely due to its own weight. It defines the key parameters that affect elongation such as the length, cross-sectional area, Young's modulus, and specific weight of the material. An equation is derived to calculate the total elongation of the bar by integrating the elongation of small sections over the length of the bar. Several example problems are then presented and solved using this equation to find the elongation of bars or wires under self-weight loading conditions.
The document is a chapter lecture summary on influence lines for structural analysis. It covers influence lines for beams, frames, girders with floor systems, and trusses. The key concepts covered include using equilibrium methods to determine influence lines for reactions, shear, and bending moment. Worked examples are provided to illustrate how to draw influence lines for specific structural elements.
it describes: 1- What is AC/AC Converter and its applilcation?
2- AC Converter in resistive and inductive load with equations
3- using phase control and Time Proportional Control
Regression analysis is a mathematical measure of the average relationship between two or more variables in terms of the original units of the data.
In regression analysis there are two types of variables. The variable whose value is influenced or is to be predicted is called dependent variable and the variable which influences the values or is used for prediction, is called independent variable.
In regression analysis independent variable is also known as regressor or predictor or explanatory variable while the dependent variable is also known as regressed or explained variable.
Lecture Notes - EEEC6430310 Electromagnetic Fields and Waves - Smith ChartAIMST University
The Smith chart is a graphical tool used to analyze high frequency circuits. It represents all possible complex impedances in terms of the reflection coefficient. Circles of constant resistance and arcs of constant reactance intersect on the chart to indicate impedance values. The chart can be used to determine impedances, reflection coefficients, standing wave ratios, and more from various circuit parameters. It provides a clever way to visualize complex impedance functions that continues to be popular for high frequency applications.
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
FUNDAMENTALS OF FLUID FLOW 3rd Edition .pdfWasswaderrick3
In this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
FUNDAMENTALS OF FLUID FLOW 3rd Edition.pdfWasswaderrick3
In this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
n this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
The document discusses key concepts related to calculus including:
- The definition of a derivative as the instantaneous rate of change of a function, obtained by taking the limit of the average rate of change as the change in x approaches 0.
- Techniques for finding derivatives including differentiation rules for basic functions.
- Relationship between a function's derivative and whether it is increasing or decreasing over an interval.
- Concepts of local/global extrema and how to analyze a function's critical points and inflection points.
- Using optimization techniques like taking derivatives to find maximum/minimum values of expressions subject to constraints.
This document discusses complex stress resulting from combinations of different loading types. It begins by introducing complex stress situations where multiple loading types like axial load, bending moment, shear, and torsion act simultaneously.
It then examines plane stress, where only stresses parallel to two axes act on an infinitesimal element. Equations are provided to transform the stress components when the element is rotated. Special cases like uniaxial stress, pure shear stress, and biaxial stress are also examined.
The document concludes by discussing principal stresses, which are the maximum and minimum normal stresses, and maximum shear stresses, which occur on planes oriented at 45 degrees to the principal planes. Equations are given to calculate these important stress
The document defines and proves theorems about the distance from a point to a line and the perpendicular bisector of a line segment. It shows that:
1) A point on the bisector of an angle is equidistant from the sides of the angle.
2) A point equidistant from the sides of an angle lies on the bisector of the angle.
3) A point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
4) A point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.
This document discusses trigonometric identities. It describes the basic types of identities such as basic identities, sum and difference identities, double angle identities, and half angle identities. It also discusses product-sum and sum-product identities. Examples of specific identities are given such as the Pythagorean identity, reciprocal identities, and even-odd identities. Methods for deriving identities from angle addition or subtraction are explained. Several examples of using identities to solve for trigonometric functions of specific angles are provided.
The document discusses properties of vector products. It defines the vector product of two vectors a and b as a × b and lists some of its key properties: a × b is perpendicular to both a and b; a × b · a = 0 and a × b · b = 0. It also discusses using the vector product to find a line perpendicular to two given lines and defines the vector product in terms of its Cartesian components.
The document is a collection of pages that repeatedly state that Bibin Chidambaranathan is an Associate Professor at RMKCET. Each page contains his name, title, and place of work.
The document is a collection of pages that repeatedly state that Bibin Chidambaranathan is an Associate Professor at RMKCET. Each page contains his name, title, and place of work.
This document presents two problems related to calculating elastic constants for mechanical structures. The first problem calculates the Poisson's ratio and modulus of elasticity for a bar where lateral strain is given. The second problem calculates the modulus of rigidity, bulk modulus, and change in volume for a cylindrical bar under hydrostatic pressure, given the longitudinal and lateral strains. Formulas for stress, strain, elastic constants, volume, and volumetric strain are provided. The values of the elastic constants and change in volume are calculated for each problem.
CR- Submanifoldsof a Nearly Hyperbolic Cosymplectic ManifoldIOSR Journals
This document discusses properties of CR-submanifolds of a nearly hyperbolic cosymplectic manifold. It begins with definitions of nearly hyperbolic cosymplectic manifolds and CR-submanifolds. The paper then obtains some results about the horizontal and vertical distributions on CR-submanifolds, including that the horizontal distribution is invariant by the structure tensor φ, while the vertical distribution is anti-invariant by φ. Lemmas are proved relating the covariant derivatives of φ on the horizontal and vertical distributions to terms involving the second fundamental form and normal connection.
This document presents the analysis of non-lifting potential flow past a thin symmetric hydrofoil using a finite difference method. The objectives are to solve the potential flow problem around a 2D hydrofoil and calculate the pressure distribution. A NACA 0012 hydrofoil is used. Governing equations and boundary conditions for the potential flow are described. A structured algebraic grid is used to generate points around the hydrofoil. Finite difference equations are derived and discretized on the grid points. The pressure coefficient is then calculated and compared to experimental results, showing good agreement.
Estimation Theory Class (Summary and Revision)Ahmad Gomaa
Summary of important theories and formulas in Estimation theory:
1) Cramer-Rao lower bound (CRLB)
2) Linear Model
3) Best Linear Unbiased Estimate (BLUE)
4) Maximum Likelihood Estimation (MLE)
5) Least Squares Estimation (LSE)
6) Bayesian Estimation and MMSE estimation
The document discusses the elongation of a bar hanging freely due to its own weight. It defines the key parameters that affect elongation such as the length, cross-sectional area, Young's modulus, and specific weight of the material. An equation is derived to calculate the total elongation of the bar by integrating the elongation of small sections over the length of the bar. Several example problems are then presented and solved using this equation to find the elongation of bars or wires under self-weight loading conditions.
The document is a chapter lecture summary on influence lines for structural analysis. It covers influence lines for beams, frames, girders with floor systems, and trusses. The key concepts covered include using equilibrium methods to determine influence lines for reactions, shear, and bending moment. Worked examples are provided to illustrate how to draw influence lines for specific structural elements.
it describes: 1- What is AC/AC Converter and its applilcation?
2- AC Converter in resistive and inductive load with equations
3- using phase control and Time Proportional Control
Regression analysis is a mathematical measure of the average relationship between two or more variables in terms of the original units of the data.
In regression analysis there are two types of variables. The variable whose value is influenced or is to be predicted is called dependent variable and the variable which influences the values or is used for prediction, is called independent variable.
In regression analysis independent variable is also known as regressor or predictor or explanatory variable while the dependent variable is also known as regressed or explained variable.
Lecture Notes - EEEC6430310 Electromagnetic Fields and Waves - Smith ChartAIMST University
The Smith chart is a graphical tool used to analyze high frequency circuits. It represents all possible complex impedances in terms of the reflection coefficient. Circles of constant resistance and arcs of constant reactance intersect on the chart to indicate impedance values. The chart can be used to determine impedances, reflection coefficients, standing wave ratios, and more from various circuit parameters. It provides a clever way to visualize complex impedance functions that continues to be popular for high frequency applications.
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
FUNDAMENTALS OF FLUID FLOW 3rd Edition .pdfWasswaderrick3
In this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
FUNDAMENTALS OF FLUID FLOW 3rd Edition.pdfWasswaderrick3
In this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
n this book we use energy conservation techniques on a differential element and derive the governing equation from which we get the Toricelli , Pouiselle , transition and turbulent flow equations all from one equation. The most important value in using the energy conservation techniques is to first get the Friction factor by solving the corresponding Navier Stoke's equation for a given geometry of pipe for laminar flow and then find the Friction factor whose value we use to derive the Toricelli governing equation, the laminar flow equation, turbulent flow and transition flow equations. We note that in Toricelli flow the length of the pipe is reduced to zero and we go ahead to derive the governing equation. We go ahead to look at the head loss equations and derive the friction factors from them. we also go ahead to look at a spherical body falling under the influence of gravity alone and we derive the governing equations and terminal velocity equations. Other phenomena are explained too
The document discusses key concepts related to calculus including:
- The definition of a derivative as the instantaneous rate of change of a function, obtained by taking the limit of the average rate of change as the change in x approaches 0.
- Techniques for finding derivatives including differentiation rules for basic functions.
- Relationship between a function's derivative and whether it is increasing or decreasing over an interval.
- Concepts of local/global extrema and how to analyze a function's critical points and inflection points.
- Using optimization techniques like taking derivatives to find maximum/minimum values of expressions subject to constraints.
This document discusses complex stress resulting from combinations of different loading types. It begins by introducing complex stress situations where multiple loading types like axial load, bending moment, shear, and torsion act simultaneously.
It then examines plane stress, where only stresses parallel to two axes act on an infinitesimal element. Equations are provided to transform the stress components when the element is rotated. Special cases like uniaxial stress, pure shear stress, and biaxial stress are also examined.
The document concludes by discussing principal stresses, which are the maximum and minimum normal stresses, and maximum shear stresses, which occur on planes oriented at 45 degrees to the principal planes. Equations are given to calculate these important stress
The document defines and proves theorems about the distance from a point to a line and the perpendicular bisector of a line segment. It shows that:
1) A point on the bisector of an angle is equidistant from the sides of the angle.
2) A point equidistant from the sides of an angle lies on the bisector of the angle.
3) A point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
4) A point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.
This document discusses trigonometric identities. It describes the basic types of identities such as basic identities, sum and difference identities, double angle identities, and half angle identities. It also discusses product-sum and sum-product identities. Examples of specific identities are given such as the Pythagorean identity, reciprocal identities, and even-odd identities. Methods for deriving identities from angle addition or subtraction are explained. Several examples of using identities to solve for trigonometric functions of specific angles are provided.
The document discusses properties of vector products. It defines the vector product of two vectors a and b as a × b and lists some of its key properties: a × b is perpendicular to both a and b; a × b · a = 0 and a × b · b = 0. It also discusses using the vector product to find a line perpendicular to two given lines and defines the vector product in terms of its Cartesian components.
The document is a collection of pages that repeatedly state that Bibin Chidambaranathan is an Associate Professor at RMKCET. Each page contains his name, title, and place of work.
The document is a collection of pages that repeatedly state that Bibin Chidambaranathan is an Associate Professor at RMKCET. Each page contains his name, title, and place of work.
The document is a collection of 116 pages that all state "BIBIN CHIDAMBARANATHAN ASSOCIATE PROFESSOR RMKCET" in large text. Each page appears to be a scanned page from a document or form with this repeated text.
The document is a collection of pages that repeatedly state that Bibin Chidambaranathan is an Associate Professor at RMKCET. Each page contains his name, title, and place of work.
The document consists of 128 pages that are all identical. On each page is the text "BIBIN CHIDAMBARANATHAN ASSOCIATE PROFESSOR RMKCET" repeated, indicating it is some type of certificate or document for Bibin Chidambaranathan identifying him as an Associate Professor at RMKCET.
The document outlines a plan for a neighborhood block party to be held on June 15th from 12pm to 4pm. The event will include food, games, and music in the park located at Main Street and Elm Avenue. Residents are asked to RSVP by June 1st so organizers can plan accordingly.
This document appears to be an exam paper for a course on Lean Six Sigma. It contains 16 questions across three parts:
- Part A contains 10 multiple choice questions worth 2 marks each, related to Lean Six Sigma concepts like customer centricity, scatter charts, Pareto analysis, failure modes, DFSS methodology, champions, organizational structures, preventative maintenance.
- Part B contains 5 questions worth 13 marks each, asking students to discuss symptoms indicating the need for Lean Six Sigma, explain the PDCA cycle and CTQ tree, define Risk Priority Number and explain the DMAIC process, discuss leadership activities and project selection in Lean Six Sigma.
- Part C contains 1 question worth 15 marks, asking students to
This document outlines the structure and content of an examination for a Product Design and Development course. It contains 3 parts (A, B, C) with multiple choice and long answer questions. Part A contains 10 short answer questions about issues to consider in project/product design, concept selection benefits, product performance estimation, concept generation approaches, product architecture establishment, industrial design needs, and design for manufacturing guidelines. Part B contains 3 long answer questions about product development processes, concept generation methods, and product architecture types and implications. Part C contains 1 long answer question about concept selection and generation methods.
This document contains a case study about Matt James, the new manager of Health-Time fitness club. He implemented stricter rules and schedules for staff to improve finances. However, morale declined as some staff quit or switched jobs. Matt is perplexed about the staff's lack of motivation and pride in the club's success. He asks for strategies to address the problem. The case study is followed by four questions analyzing the motivational issues using Maslow's hierarchy, expectancy theory, and equity theory. An optional question asks to justify if concentration of power or delegation of authority is better for an organization.
This document contains a question paper for an entrepreneurship development exam with three parts. Part A contains 10 short answer questions about topics like defining entrepreneurship, characteristics of entrepreneurs, motivation, market research, project reports, and more. Part B contains 5 long answer questions about types of entrepreneurs, entrepreneurial growth factors, stress management techniques, setting up a business, sources of financial support, and more. Part C contains 1 long answer question about developing a business plan or discussing policy support for small industries. The exam covers a range of entrepreneurship topics and tests students' understanding through definitions, explanations, discussions, and developing a business plan.
This document contains a question paper for an entrepreneurship development exam with three parts. Part A contains 10 short answer questions about topics like defining entrepreneurship, characteristics of entrepreneurs, motivation, market research, project reports, and more. Part B contains 5 long answer questions about types of entrepreneurs, entrepreneurial growth factors, stress management techniques, setting up a business, sources of financial support, and more. Part C contains 1 long answer question about developing a business plan or discussing policy support for small industries. The exam covers a range of entrepreneurship topics and tests students' understanding through definitions, explanations, discussions, and developing a business plan.
This document contains details regarding a mechanical engineering examination, including questions on topics like process planning, cost estimation, machine selection, jigs and fixtures design, and manufacturing processes. It provides multiple choice and numerical questions to calculate costs, times, and parameters for various manufacturing steps like welding, turning, shaping and upsetting.
This document provides information about an examination for a power plant engineering course, including questions that assess knowledge of various power plant components, systems, and principles. It begins with general multiple choice and short answer questions about different power plant types. Longer questions then require explaining the working principles and components of specific power plant technologies like circulating fluidized bed combustors, supercritical boilers, diesel power plants, gas turbines, nuclear reactors, wind turbines, and solar power plants. The document concludes with numerical problems involving calculating costs and efficiencies for different power generation scenarios.
This document contains questions for an exam on the topic of Mechatronics. It includes multiple choice and descriptive questions in three parts:
Part A contains 10 multiple choice questions worth 2 marks each, covering topics like measurement systems, potentiometers, microprocessors, interfacing, PLCs and stepper motors.
Part B contains 5 questions worth 13 marks each, with choices between two questions on topics like sensors, microprocessor architecture, interfacing, PLC architecture and applications of stepper motors.
Part C contains one 15 mark question providing a circuit design problem using a PLC or describing a car engine management system using a mechatronic approach.
This document contains questions that will be asked in an examination for a Mechanical Engineering course on hydraulics and pneumatics. It includes questions in three parts - Part A contains 10 short answer questions worth 2 marks each about fluid power applications, components, and concepts. Part B contains 5 longer answer questions worth 13 marks each about hydraulic pumps, motors, circuits and accumulators. Part C contains 1 question worth 15 marks asking to either design a fluid power circuit for a drilling machine or develop a pneumatic circuit using cascade method for multiple cylinders.
This document contains details regarding an examination for a heat and mass transfer course, including:
1) The exam has 3 parts consisting of short answer, long answer, and case study questions worth 20, 65, and 15 marks respectively.
2) Sample questions assess topics like fin analysis, boundary layers, heat exchanger design, phase change processes, and radiation heat transfer.
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How to Manage Reception Report in Odoo 17Celine George
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
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THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
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A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
3. Thick cylinders
• If the ratio of thickness to internal diameter of a cylindrical shell is less than about
1/20, the cylindrical shell is known as thin cylinders.
• The hoop and longitudinal stresses are constant over the thickness and the radial
stress is small and can be neglected.
• If the ratio of thickness to internal diameter is more than 1/20, then cylindrical
shell is known as thick cylinders.
• The hoop stress in case of a thick cylinder will not be uniform across the thickness.
• Actually, the hoop stress will vary from a maximum value at the inner
circumference to a minimum value at the outer circumference.
3 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
4. STRESSES IN A THICK CYLINDRICAL SHELL
Fig. shows a thick cylinder subjected to an internal fluid pressure.
4 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
5. Let
𝑟2 = External radius of the cylinder,
𝑟1 = Internal radius of the cylinder, and
𝐿 = Length of cylinder.
Consider an elementary ring of the cylinder of radius 𝑥 and thickness 𝑑𝑥 as shown in Fig
5 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
6. Let 𝑝𝑥 = Radial pressure on the inner surface of the ring
𝑝𝑥 + 𝑑𝑝𝑥 = Radial pressure on the outer surface of the ring
𝜎𝑥 = Hoop stress induced in the ring.
Take a longitudinal section 𝑥 − 𝑥 and consider the equilibrium of half of the ring of Fig.
6 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
8. Equating the resisting force to the bursting force, we get
𝜎𝑥 × 2𝑑𝑥. 𝐿 = 2𝐿[𝑝𝑥 𝑑𝑥 + 𝑥𝑑𝑝𝑥]
𝜎𝑥 = −𝑝𝑥 − 𝑥
𝑑𝑝𝑥
𝑑𝑥
The longitudinal strain at any point in the section is constant and is independent of the radius. This
means that cross sections remain plane after straining and this is true for sections, remote from any end
fixing. As longitudinal strain is constant, hence longitudinal stress will also be constant.
8 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
9. Let
𝜎𝑥 = 𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠
Hence at any point at a distance 𝑥 from the centre, three principal stresses are acting
They are:
(i) the radial compressive stress, 𝑃𝑥
(ii) the hoop (or circumferential) tensile stress, 𝜎𝑥
(iii) the longitudinal tensile stress 𝜎2
9 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
10. The longitudinal strain (𝑒2) at this point is given by,
𝑒2 =
𝜎2
𝐸
−
𝜇𝜎𝑥
𝐸
+
𝜇 𝑝𝑥
𝐸
But longitudinal strain is constant.
𝜎2
𝐸
−
𝜇𝜎𝑥
𝐸
+
𝜇 𝑝𝑥
𝐸
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
But 𝜎2 is also constant, and for the material of the cylinder 𝐸 and μ are constant.
𝜎𝑥 − 𝑃𝑥 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜎𝑥 − 𝑃𝑥 = 2𝑎
10 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
11. where a is constant
𝜎𝑥 = 𝑃𝑥 + 2𝑎
Equating the two values of 𝜎𝑥 given by equations (iii) and (iv), we get
𝑃𝑥 + 2𝑎 = −𝑝𝑥 − 𝑥
𝑑𝑝𝑥
𝑑𝑥
𝑥
𝑑𝑝𝑥
𝑑𝑥
= −𝑝𝑥 − 𝑝𝑥 − 2𝑎
𝑥
𝑑𝑝𝑥
𝑑𝑥
= −𝑝𝑥 − 𝑝𝑥 − 2𝑎
11 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
13. Integrating the above equation, we get
log𝑒(𝑝𝑥 + 𝑎) = − 2 log𝑒 𝑥 + log𝑒 𝑏
Where
log𝑒 𝑏 is a constant of integration
The above equation can also be written as
log𝑒(𝑝𝑥 + 𝑎) = − log𝑒 𝑥2
+ log𝑒 𝑏
log𝑒(𝑝𝑥 + 𝑎) = − log𝑒
𝑏
𝑥2
13 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
14. 𝑝𝑥 + 𝑎 =
𝑏
𝑥2
𝑝𝑥 =
𝑏
𝑥2
− 𝑎
This equation gives the radial pressure 𝑝𝑥
Substituting the values of 𝑃𝑥 in equation (iv), we get
𝜎𝑥 =
𝑏
𝑥2
− 𝑎 + 2𝑎
𝜎𝑥 =
𝑏
𝑥2
+ 𝑎
14 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
15. This equation gives the hoop stress at any radius x. These two equations are called Lame's
equations. The constants 'a' and 'b' are obtained from boundary conditions, which are:
(i) at 𝑥 = 𝑟1, 𝑝𝑥 = 𝑝0 or the pressure of fluid inside the cylinder and
(ii) at 𝑥 = 𝑟2, 𝑝𝑥 = 0 or atmosphere pressure.
After knowing the values of 'a' and 'b', the hoop stress can be calculated at any radius.
15 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
16. Alternate Method for finding stresses in a thick-
cylinder
In case of thick cylinders, due to internal fluid pressure, the three principal stresses acting at a point
are:
(i) radial pressure (p) which is compressive,
(ii) circumferential stress or hoop stress (𝜎1) which is tensile, and
(iii) longitudinal stress (𝜎2), which is also tensile.
Let
𝑒1 = 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛,
𝑒2 = 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛, 𝑎𝑛𝑑
𝑒𝑟 = 𝑟𝑎𝑑𝑖𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
16 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
17. In case of thick-cylinder, it may be assumed that longitudinal strain (𝑒2) is constant,
which means that cross-section remain plane after straining.
Consider a circular ring of radius rand thickness '𝑑𝑟'. Due to internal fluid pressure, let
the radius r increases to (𝑟 + 𝑢) and increase in the thickness 𝑑𝑟 be 𝑑𝑢.
Initial radius = r
whereas final radius = (r + u)
𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 =
𝐹𝑖𝑛𝑎𝑙 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 − 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒
17 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
20. The circumferential strain, longitudinal strain and radial strains in terms of stresses
and Poisson's ratio are also given by,
𝑒1 =
𝜎1
𝐸
−
𝜇𝜎2
𝐸
−
−𝜇 𝑝
𝐸
(P is compressive)
𝑒1 =
𝜎1
𝐸
−
𝜇𝜎2
𝐸
+
𝜇 𝑝
𝐸
20 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
21. 𝑒2 =
𝜎2
𝐸
−
𝜇𝜎1
𝐸
+
𝜇 𝑝
𝐸
𝑒𝑟 =
−𝑝
𝐸
−
𝜇𝜎1
𝐸
+
𝜇𝜎2
𝐸
Equating the two values of circumferential strain (𝑒1) given by equations (i) and (iii).
Also equate the two values of radial strain (𝑒𝑟) given by equations (ii) and (v), we get
𝑢
𝑟
=
𝜎1
𝐸
−
𝜇𝜎2
𝐸
+
𝜇 𝑝
𝐸
21 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
22. 𝑑𝑢
𝑑𝑟
=
−𝑝
𝐸
−
𝜇𝜎1
𝐸
+
𝜇𝜎2
𝐸
Let us first eliminate 'u' from equations (vi) and (vii). From equation (vi),
𝑢 = [
𝜎1
𝐸
−
𝜇𝜎2
𝐸
+
𝜇 𝑝
𝐸
] × 𝑟
Differentiating the above equation with respect to r, we get
𝑑𝑢
𝑑𝑟
=
𝜎1
𝐸
−
𝜇𝜎2
𝐸
+
𝜇 𝑝
𝐸
+
𝑟
𝐸
(
𝑑𝜎1
𝑑𝑟
−
𝜇 𝑑𝜎2
𝑑𝑟
+
𝜇 𝑑𝑝
𝑑𝑟
)
22 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
25. To find the value of 𝜎1 in terms of p, the value of
𝑑𝜎2
𝑑𝑟
must be substituted in the above
equation (ix).
Longitudinal strain (𝑒2) is given by equation (iv) as
𝑒2 =
𝜎2
𝐸
−
𝜇𝜎1
𝐸
+
𝜇 𝑝
𝐸
Since longitudinal strain (𝑒2) is assumed constant. Its differentiation with respect to r will be
zero. D1fferentiating the above equation with respect to 'r', we get
0 =
1
𝐸
[
𝑑𝜎2
𝑑𝑟
−
𝜇𝑑𝜎1
𝑑𝑟
+
𝜇 𝑑𝑝
𝑑𝑟
]
25 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
26. 0 =
𝑑𝜎2
𝑑𝑟
−
𝜇𝑑𝜎1
𝑑𝑟
+
𝜇 𝑑𝑝
𝑑𝑟
𝑑𝜎2
𝑑𝑟
=
𝜇𝑑𝜎1
𝑑𝑟
−
𝜇 𝑑𝑝
𝑑𝑟
Substituting the value of
𝑑𝜎2
𝑑𝑟
in equation (ix), we get
0 = (𝑝 + 𝜎1)(1 + 𝜇) + 𝑟(
𝑑𝜎1
𝑑𝑟
−
𝜇 𝑑𝜎2
𝑑𝑟
+
𝜇 𝑑𝑝
𝑑𝑟
)
26 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
29. For the equilibrium of the element, resolving the forces acting on the element in the
radial direction we get
0 = 𝑝 + 𝑑𝑝 𝑟 + 𝑑𝑟 𝛿𝜃 + 2𝜎1 × 𝑑𝑟 × 𝑠𝑖𝑛
𝛿𝜃
2
As 𝛿𝜃 is small, hence 𝑠𝑖𝑛
𝛿𝜃
2
=
𝛿𝜃
2
Now the above equation becomes
0 = 𝑝 + 𝑑𝑝 𝑟 + 𝑑𝑟 𝛿𝜃 + 2𝜎1 × 𝑑𝑟 × 𝑠𝑖𝑛
𝛿𝜃
2
29 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
34. In equations (xii) and (xi), the two unknowns are 𝜎1 and 𝑝. Let us find the value of
𝜎1 from equation (xii) and substitute this value in equation (xi).
𝑝 + (2𝑎 + 𝑝) = −𝑟
𝑑𝑝
𝑑𝑟
2𝑎 + 2𝑝 = −𝑟
𝑑𝑝
𝑑𝑟
2𝑝 + 𝑟
𝑑𝑝
𝑑𝑟
= −2𝑎
34 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
36. 𝑝𝑟2
= 𝑎𝑟2
+ 𝑏
𝑝 =
−𝑎𝑟2
𝑟2
+
𝑏
𝑟2
𝑝 =
𝑏
𝑟2
− 𝑎
The above equation gives the radial pressure at radius 'r'. This equation is same as
equation. To find the value of 𝜎1 at any radius, substitute this value of p in equation (xiii).
36 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
37. 𝜎1 −
𝑏
𝑟2
− 𝑎 = 2𝑎
𝜎1 −
𝑏
𝑟2
+ 𝑎 = 2𝑎
𝜎1 =
𝑏
𝑟2
+ 2𝑎 − 𝑎
𝜎1 =
𝑏
𝑟2
+ 𝑎
The above equation gives the circumferential stress (or hoop stress) at any radius r.
This equation is same as equation
37 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
41. Problem 1.
Determine the maximum and minimum hoop stress across the section of a pipe of 400 mm
internal diameter and 100 mm thick, when the pipe contains a fluid at a pressure of 8 𝑁/𝑚𝑚2
.
Also sketch the radial pressure distribution and hoop stress distribution across the section.
Internal dia. = 400 mm
Internal radius = 200 mm
Thickness = 100 mm
External dia. = 600 mm
External radius = 300 mm
41 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
42. Fluid pressure, p = 8 𝑁/𝑚𝑚2
The radial pressure (px) is given by equation
𝑝𝑥 =
𝑏
𝑟2
− 𝑎
Now apply the boundary conditions to the above equations. The boundary
conditions
(i) at 𝑥 = 𝑟1, 𝑝𝑥 = 𝑝0 or the pressure of fluid inside the cylinder and
(ii) at 𝑥 = 𝑟2, 𝑝𝑥 = 0 or atmosphere pressure.
Substituting these boundary conditions in equation (i), we get
42 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
48. 𝜎300 =
576000
3002
+ 6.4
𝜎300 = 6.4 + 6.4
𝜎300 = 12.8 𝑁/𝑚𝑚2
Fig. 18.3 shows the radial pressure distribution and hoop stress distribution across the
section. AB is taken a horizontal line. AC= 8 𝑁/𝑚𝑚2
. The variation between B and C is
parabolic. The curve BC shows the variation of radial pressure across AB.
48 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
49. The curve DE which is also parabolic, shows the variation of hoop stress across AB.
Values BD = 12.8 𝑁/𝑚𝑚2
and AE = 20.8 𝑁/𝑚𝑚2
. The radial pressure is compressive whereas the
hoop stress is tensile.
49 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
50. Problem 2.
Find the thickness of metal necessary for a cylindrical shell of internal diameter 160 mm to
withstand an internal pressure of 8 𝑁/𝑚𝑚2
. The maximum hoop stress in the section is not to
exceed 35 𝑁/𝑚𝑚2
.
Internal dia. = 160 mm
Internal radius =80 mm
Internal pressure = 8 𝑁/𝑚𝑚2
This means at x = 80 mm,
𝑝𝑥 = 8 𝑁/𝑚𝑚2
50 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
51. Maximum hoop stress
𝜎𝑥 = 35 𝑁/𝑚𝑚2
The maximum hoop stress is at the inner radius of the shell.
Let
𝑟2 = External radius.
The radial pressure and hoop stress at any radius 𝑥 are given by equations
𝑝𝑥 =
𝑏
𝑟2
− 𝑎
𝜎𝑥 =
𝑏
𝑥2
+ 𝑎
51 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
52. Now apply the boundary conditions to the above equations. The boundary conditions
(i) at 𝑥 = 𝑟1, 𝑝𝑥 = 𝑝0 or the pressure of fluid inside the cylinder and
(ii) at 𝑥 = 𝑟2, 𝑝𝑥 = 0 or atmosphere pressure.
Substituting these boundary conditions in equation (i), we get
8 =
𝑏
802
− 𝑎
8 =
𝑏
6400
− 𝑎
52 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
53. Subtracting equation (iii) from equation (ii), we get
35 =
𝑏
802
+ 𝑎
35 =
𝑏
6400
+ 𝑎
Subtracting equation (iii) from equation (iv), we get
27 = 2𝑎
𝑎 = 13.5
53 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
54. Substituting the value of a in equation (iii), we get
8 =
𝑏
6400
− 13.5
𝑏 = 21.5 × 6400
Substituting the values of 'a' and 'b' in equation (i),
𝑝𝑥 =
21.5 × 6400
𝑥2
− 13.5
54 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7
55. But at the outer surface, the pressure is zero. Hence at 𝑥 = 𝑟2, 𝑃𝑥 = 0.
Substituting these values in the above equation, we get
0 =
21.5 × 6400
𝑟1
2 − 13.5
𝑟2
2
=
21.5 × 6400
13.5
𝑟2 =
21.5 × 6400
13.5
55 BIBIN CHIDAMBARANATHAN, ASP/MECH, RMKCET 5/7