1. The document discusses stress transformation equations which allow calculation of stresses on inclined planes from known normal and shear stresses.
2. It provides the equations to calculate normal (σx1, σy1) and shear (τx1y1) stresses on a plane rotated by an angle θ from the original plane with known σx, σy, and τxy stresses.
3. The principal stresses σ1 and σ2, which represent maximum and minimum normal stresses, can be calculated from the original stresses using the transformation equations. The principal planes where these stresses act have no shear stresses.
Mohr's circle is a graphical representation of the transformation equations for plane stress. It allows visualization of normal and shear stresses on inclined planes at a point in a stressed body. Using Mohr's circle, one can calculate principal stresses, maximum shear stresses, and stresses on inclined planes. The procedure involves plotting the initial stress state as two points A and B on a circle, then rotating the coordinate system to determine stresses under different inclinations. Two examples demonstrate using Mohr's circle to find principal stresses, maximum shear stresses, and stresses on a plane inclined at 30 degrees.
1. The document discusses concepts in mechanics of materials including stress, strain, elastic deformation, stress-strain curves, shear stress, normal stress, Poisson's ratio, and elastic strain energy.
2. Various equations are presented for calculating stress, strain, elastic moduli like Young's modulus, shear modulus, and bulk modulus.
3. Examples are provided to demonstrate relationships between stress and strain components, normal and shear strains, and derivation of equations for elastic moduli.
This document discusses strain rosette analysis using strain gauges. It begins by introducing strain gauges and how they work by converting mechanical displacement into electrical resistance changes. It then discusses the different types of strain gauges and rosette configurations. The key aspects covered are:
- Strain rosettes use multiple strain gauges oriented in different directions to measure normal strains and shear strains on a surface.
- Three common rosette types are tee, rectangular, and delta configurations with 2-3 gauges.
- Proper numbering and orientation of gauges is important for calculating principal strains and stresses from the measurements.
- Equations relate the measured strain values to the principal strains and maximum shear strain based
1. When a force is applied to a body, it causes the body to deform or change shape. This deformation is called strain. Direct stress is calculated as the applied force divided by the cross-sectional area.
2. Materials deform both elastically and plastically when stressed. Elastic deformation is reversible but plastic deformation causes a permanent change in shape. Hooke's law describes the linear elastic behavior of many materials, where stress is directly proportional to strain up to the elastic limit.
3. Thermal expansion and contraction can induce stress in materials as temperature changes unless deformation is unconstrained. The total strain is the sum of strain due to stress and strain due to temperature changes.
The document discusses stress and strain in materials. It introduces the key concepts of normal stress, shear stress, bearing stress, and thermal stress. Normal stress acts perpendicular to a cross-section, shear stress acts tangentially, and bearing stress occurs at contact points. The relationships between stress, strain, elastic modulus, and Poisson's ratio are explained. Methods for calculating stress and strain in axial loading, torsion, bending and combined loading are presented through examples. The stress-strain diagram is discussed to show material properties like yield strength and ductility.
This document provides an overview of topics related to simple stresses and strains, including:
- Types of stresses and strains such as tensile, compressive, direct stress, and direct strain.
- Hooke's law and how stress is proportional to strain below the material's yield point.
- Stress-strain diagrams and key points such as the elastic region, yield point, and fracture point.
- Definitions of terms like working stress, factor of safety, Poisson's ratio, and elastic moduli.
- Examples of problems calculating stresses, strains, extensions, and deformations of simple structural members under various loads.
Mohr's circle is a graphical representation of the transformation equations for plane stress. It allows visualization of normal and shear stresses on inclined planes at a point in a stressed body. Using Mohr's circle, one can calculate principal stresses, maximum shear stresses, and stresses on inclined planes. The procedure involves plotting the initial stress state as two points A and B on a circle, then rotating the coordinate system to determine stresses under different inclinations. Two examples demonstrate using Mohr's circle to find principal stresses, maximum shear stresses, and stresses on a plane inclined at 30 degrees.
1. The document discusses concepts in mechanics of materials including stress, strain, elastic deformation, stress-strain curves, shear stress, normal stress, Poisson's ratio, and elastic strain energy.
2. Various equations are presented for calculating stress, strain, elastic moduli like Young's modulus, shear modulus, and bulk modulus.
3. Examples are provided to demonstrate relationships between stress and strain components, normal and shear strains, and derivation of equations for elastic moduli.
This document discusses strain rosette analysis using strain gauges. It begins by introducing strain gauges and how they work by converting mechanical displacement into electrical resistance changes. It then discusses the different types of strain gauges and rosette configurations. The key aspects covered are:
- Strain rosettes use multiple strain gauges oriented in different directions to measure normal strains and shear strains on a surface.
- Three common rosette types are tee, rectangular, and delta configurations with 2-3 gauges.
- Proper numbering and orientation of gauges is important for calculating principal strains and stresses from the measurements.
- Equations relate the measured strain values to the principal strains and maximum shear strain based
1. When a force is applied to a body, it causes the body to deform or change shape. This deformation is called strain. Direct stress is calculated as the applied force divided by the cross-sectional area.
2. Materials deform both elastically and plastically when stressed. Elastic deformation is reversible but plastic deformation causes a permanent change in shape. Hooke's law describes the linear elastic behavior of many materials, where stress is directly proportional to strain up to the elastic limit.
3. Thermal expansion and contraction can induce stress in materials as temperature changes unless deformation is unconstrained. The total strain is the sum of strain due to stress and strain due to temperature changes.
The document discusses stress and strain in materials. It introduces the key concepts of normal stress, shear stress, bearing stress, and thermal stress. Normal stress acts perpendicular to a cross-section, shear stress acts tangentially, and bearing stress occurs at contact points. The relationships between stress, strain, elastic modulus, and Poisson's ratio are explained. Methods for calculating stress and strain in axial loading, torsion, bending and combined loading are presented through examples. The stress-strain diagram is discussed to show material properties like yield strength and ductility.
This document provides an overview of topics related to simple stresses and strains, including:
- Types of stresses and strains such as tensile, compressive, direct stress, and direct strain.
- Hooke's law and how stress is proportional to strain below the material's yield point.
- Stress-strain diagrams and key points such as the elastic region, yield point, and fracture point.
- Definitions of terms like working stress, factor of safety, Poisson's ratio, and elastic moduli.
- Examples of problems calculating stresses, strains, extensions, and deformations of simple structural members under various loads.
This document provides an introduction to strength of materials (SOM). It defines key terms like strength, stiffness, stability, and durability. It discusses the basic problem in SOM as developing methods to design structural elements that consider strength, stiffness, stability, and economy. It also outlines the main hypotheses in SOM, including the material being continuous, homogeneous, and isotropic. It then discusses different types of stresses like tensile, compressive, and shear stresses. It provides stress-strain curves for ductile materials and defines modulus of elasticity. Examples of calculating stresses and strains in structural elements are also provided.
Strength of Materials Lecture - 2
Elastic stress and strain of materials (stress-strain diagram)
Mehran University of Engineering and Technology.
Department of Mechanical Engineering.
This document presents explicit analytical solutions for pressure across oblique shock and expansion waves in supersonic flow. It begins by introducing the need for explicit pressure-deflection solutions in solving aerodynamic problems. It then presents:
1) Exact explicit solutions for pressure coefficient and ratio across weak and strong oblique shock waves as functions of deflection angle.
2) Third-order accurate explicit unitary solutions for pressure coefficient and ratio across oblique shocks and expansions as functions of deflection angle.
3) Numerical validation showing good agreement of the new explicit solutions with exact solutions for a range of Mach numbers and deflection angles.
The document discusses stresses in beams, including:
1. Pure bending causes zero shear force, resulting in a constant bending moment along the beam.
2. A positively curved beam under bending moment develops compression on top and tension on bottom, with the neutral axis in between with zero stress.
3. Shear stresses in beams are highest at the center of rectangular beams and at the edge of circular beams.
4. Slope, deflection, and radius of curvature are related through beam equations involving the bending moment and moment of inertia.
This document provides an overview of fundamental mechanical engineering concepts including stress, strain, Hooke's law, stress-strain diagrams, elastic constants, and mechanical properties. It defines stress as force per unit area and strain as the deformation of a material from stress. Hooke's law states that stress is directly proportional to strain within the elastic limit. Stress-strain diagrams are presented for ductile and brittle materials. Key elastic constants like Young's modulus, shear modulus, and Poisson's ratio are defined along with their relationships. Mechanical properties of materials like elasticity, plasticity, ductility, strength, brittleness, toughness, hardness, and stiffness are also summarized.
The document provides an introduction to mechanics of deformable solids. It defines stress as force per unit area and distinguishes between normal and shear stresses. Normal stresses are stresses acting perpendicular to a surface, and can be tensile or compressive. Shear stresses act parallel to a surface. The general state of stress at a point involves six independent stress components - normal stresses on three perpendicular planes and shear stresses on those planes. Notation for stresses depends on the coordinate system used.
1) The document discusses fluid kinematics, which deals with the motion of fluids without considering the forces that create motion. It covers topics like velocity fields, acceleration fields, control volumes, and flow visualization techniques.
2) There are two main descriptions of fluid motion - Lagrangian, which follows individual particles, and Eulerian, which observes the flow at fixed points in space. Most practical analysis uses the Eulerian description.
3) The Reynolds Transport Theorem allows equations written for a fluid system to be applied to a fixed control volume, which is useful for analyzing forces on objects in a flow. It relates the time rate of change of an extensive property within the control volume to surface fluxes and the property accumulation.
Here are the key steps to solve this problem:
1) Draw a free body diagram of each block, showing all external forces.
2) Write the equation of motion for each block in the x and y directions: ΣFx = max, ΣFy = may
3) The tension in each cable will be the same. Substitute this into the equations of motion.
4) Solve the equations simultaneously to find the acceleration and tension.
The acceleration and tension can be determined by setting up and solving the simultaneous equations of motion for each block based on Newton's 2nd law. Friction and the coefficient of kinetic friction must be accounted for between block C and the horizontal surface.
1. Hooke's law states that the stress and strain of a material are proportional for small deformations.
2. Young's modulus is a measure of the stiffness of a material and is defined as the ratio of tensile or compressive stress to longitudinal strain.
3. Shear modulus is defined as the ratio of shearing stress to shearing strain and measures a material's resistance to deformation via shear forces.
The document discusses Deepak's academic and professional background, including an MBA from IE Business School in Spain and experience founding perfectbazaar.com. It also provides an overview of the topics to be covered in the Strength of Materials course, such as stresses, strains, Hooke's law, and analysis of bars with varying cross-sections. The grading policy and syllabus are outlined which divide the course into 5 units covering various strength of materials concepts.
Dr. Salah Uddin teaches about relationships between elastic constants, including:
1. Young's modulus describes the proportionality between stress and strain in a simple tension or compression test.
2. Bulk modulus describes the ratio of direct stress to volumetric strain for mutually perpendicular, equal stresses.
3. Shear modulus describes the linear relationship between shear stress and shear strain.
4. Poisson's ratio describes the ratio of lateral to axial strain.
5. The elastic constants are related through equations involving Young's modulus, shear modulus, bulk modulus, and Poisson's ratio.
1.3 Stress & Strain Relationship of Hooke’s Law.pptVanathisekar2
Stress refers to external forces applied to a material, while strain refers to the deformation or change in shape of the material resulting from those stresses. Hooke's law states that within the elastic limit, the amount of strain produced is directly proportional to the stress applied. Different moduli describe the relationship between stress and strain, including Young's modulus, the bulk modulus, and the shear modulus. Stress and strain can be longitudinal, relating to changes in length, or transverse, relating to changes in width or thickness. The elastic limit is the maximum stress a material can withstand without permanent deformation, after which plastic deformation or fracture may occur.
This document provides an overview of fatigue failure resulting from variable loading. It discusses three main fatigue life methods: the stress-life method, strain-life method, and fracture mechanics method. The stress-life method is based on stress levels and is most commonly used for high-cycle fatigue predictions, while the strain-life method considers localized plastic deformation and is better for low-cycle applications. The fracture mechanics method assumes a crack is already present and is used to predict crack growth. Key concepts discussed include fatigue testing methods, S-N diagrams, endurance limits, stress concentrations, and cumulative damage.
Thermal stresses and strains occur when temperature changes cause materials to expand or contract. Thermal strain is proportional to the temperature change. Uneven heating can induce stresses if expansion is constrained. Thermal stresses are analyzed by considering both the thermal strain and any mechanical stresses from applied loads. Bars with non-uniform cross sections experience varying thermal stresses due to differences in expansion rates. Composite bars made of different materials also experience thermal stresses at their interface as the materials expand unequally with temperature changes. Thermal stresses can be calculated using equations that relate stress, strain, temperature change, elastic modulus, and coefficients of thermal expansion for the materials.
Mohr's circle is a graphical representation that illustrates the relationships between normal and shear stresses or strains at a point. It shows the two principal stresses or strains and the maximum shear stress or strain. The circle is centered at the average stress or strain and has a radius equal to the maximum shear value. Mohr's circle can be used to determine principal stresses/strains and directions, transform between stress/strain systems, and visualize how stresses/strains change with rotation. It remains a useful tool for engineers despite the availability of calculators.
This document summarizes the moment-area method for calculating deflections in beams. It discusses how the bending moment diagram can be divided into areas that correspond to rotations of the elastic curve. The sum of these areas multiplied by the distance to the centroid gives the tangential deviation, which can be used to determine the deflection. The method is applicable to statically indeterminate beams using superposition. Boundary conditions and how to handle different support types are also covered.
The document discusses plasticity theory and yield criteria. It introduces Hooke's law and its limitations under large strains. Generalized Hooke's law is presented for isotropic and anisotropic materials. Common stress-strain curves are shown including elastic-plastic and strain hardening responses. Several yield criteria are covered, including maximum principal stress, Tresca, and von Mises criteria. The von Mises criterion uses a second invariant of stress to predict yielding of ductile materials. An example compares predictions of yielding using Tresca and von Mises criteria for a given stress state in aluminum.
This document provides information about the Solid Mechanics course ME 302 taught by Dr. Nirmal Baran Hui at NIT Durgapur in West Bengal, India. It lists four required textbooks for the course and provides a detailed syllabus covering topics like stress, strain, elasticity, bending, deflection, columns, torsion, pressure vessels, combined loadings, springs, and failure theories. The document also includes examples of lecture content on stress analysis, stresses on oblique planes, and material subjected to pure shear.
Mohr circle (Complete Soil Mech. Undestanding Pakage: ABHAY)Abhay Kumar
This document discusses plane stress and strain transformations. It introduces concepts such as principal stresses/strains, maximum shear, and Mohr's circle. Mohr's circle can represent the state of stress or strain at a point on a plane. It allows determination of stresses/strains on any plane from knowledge of three independent quantities (e.g. sx, sy, txy). Strain gauges and rosettes are also discussed as tools to measure strain.
This document defines and describes different types of stresses and strains. Stress is the internal force per unit area that opposes deformation when a load is applied to a body. Strain is the deformation or change in length per original length of a body under an applied load. The types of stresses discussed include axial/direct stresses (tensile and compressive), shear stresses, bending stresses, combined stresses, and thermal stresses. Combined stress results from a combination of direct and indirect stresses, such as bending, torsional, and thermal stresses. Thermal stress is caused by changes in temperature and depends on the material's coefficient of thermal expansion.
1. The document discusses transformation equations that relate stresses on an inclined plane to stresses on a normal plane. The normal and shear stresses on the inclined plane can be determined by resolving the forces into components parallel and perpendicular to the inclined plane.
2. Principal stresses are the maximum and minimum normal stresses, which occur on principal planes where the shear stresses are zero. The principal stresses and planes can be determined by differentiating the transformation equations.
3. Maximum shear stress occurs at 45° to the normal plane and is equal to half the difference between the normal stresses.
1) Advanced metal forming techniques involve plastic deformation of metal crystals beyond their elastic limit.
2) Yield criteria determine which combination of multi-axial stresses will cause yielding, such as Tresca and von-Mises criteria relating to maximum shear stresses.
3) Mohr's circle provides a graphical representation of the state of stress at a point, showing normal and shear stresses on planes of all orientations. It can be used to determine principal stresses and maximum shear stress.
This document provides an introduction to strength of materials (SOM). It defines key terms like strength, stiffness, stability, and durability. It discusses the basic problem in SOM as developing methods to design structural elements that consider strength, stiffness, stability, and economy. It also outlines the main hypotheses in SOM, including the material being continuous, homogeneous, and isotropic. It then discusses different types of stresses like tensile, compressive, and shear stresses. It provides stress-strain curves for ductile materials and defines modulus of elasticity. Examples of calculating stresses and strains in structural elements are also provided.
Strength of Materials Lecture - 2
Elastic stress and strain of materials (stress-strain diagram)
Mehran University of Engineering and Technology.
Department of Mechanical Engineering.
This document presents explicit analytical solutions for pressure across oblique shock and expansion waves in supersonic flow. It begins by introducing the need for explicit pressure-deflection solutions in solving aerodynamic problems. It then presents:
1) Exact explicit solutions for pressure coefficient and ratio across weak and strong oblique shock waves as functions of deflection angle.
2) Third-order accurate explicit unitary solutions for pressure coefficient and ratio across oblique shocks and expansions as functions of deflection angle.
3) Numerical validation showing good agreement of the new explicit solutions with exact solutions for a range of Mach numbers and deflection angles.
The document discusses stresses in beams, including:
1. Pure bending causes zero shear force, resulting in a constant bending moment along the beam.
2. A positively curved beam under bending moment develops compression on top and tension on bottom, with the neutral axis in between with zero stress.
3. Shear stresses in beams are highest at the center of rectangular beams and at the edge of circular beams.
4. Slope, deflection, and radius of curvature are related through beam equations involving the bending moment and moment of inertia.
This document provides an overview of fundamental mechanical engineering concepts including stress, strain, Hooke's law, stress-strain diagrams, elastic constants, and mechanical properties. It defines stress as force per unit area and strain as the deformation of a material from stress. Hooke's law states that stress is directly proportional to strain within the elastic limit. Stress-strain diagrams are presented for ductile and brittle materials. Key elastic constants like Young's modulus, shear modulus, and Poisson's ratio are defined along with their relationships. Mechanical properties of materials like elasticity, plasticity, ductility, strength, brittleness, toughness, hardness, and stiffness are also summarized.
The document provides an introduction to mechanics of deformable solids. It defines stress as force per unit area and distinguishes between normal and shear stresses. Normal stresses are stresses acting perpendicular to a surface, and can be tensile or compressive. Shear stresses act parallel to a surface. The general state of stress at a point involves six independent stress components - normal stresses on three perpendicular planes and shear stresses on those planes. Notation for stresses depends on the coordinate system used.
1) The document discusses fluid kinematics, which deals with the motion of fluids without considering the forces that create motion. It covers topics like velocity fields, acceleration fields, control volumes, and flow visualization techniques.
2) There are two main descriptions of fluid motion - Lagrangian, which follows individual particles, and Eulerian, which observes the flow at fixed points in space. Most practical analysis uses the Eulerian description.
3) The Reynolds Transport Theorem allows equations written for a fluid system to be applied to a fixed control volume, which is useful for analyzing forces on objects in a flow. It relates the time rate of change of an extensive property within the control volume to surface fluxes and the property accumulation.
Here are the key steps to solve this problem:
1) Draw a free body diagram of each block, showing all external forces.
2) Write the equation of motion for each block in the x and y directions: ΣFx = max, ΣFy = may
3) The tension in each cable will be the same. Substitute this into the equations of motion.
4) Solve the equations simultaneously to find the acceleration and tension.
The acceleration and tension can be determined by setting up and solving the simultaneous equations of motion for each block based on Newton's 2nd law. Friction and the coefficient of kinetic friction must be accounted for between block C and the horizontal surface.
1. Hooke's law states that the stress and strain of a material are proportional for small deformations.
2. Young's modulus is a measure of the stiffness of a material and is defined as the ratio of tensile or compressive stress to longitudinal strain.
3. Shear modulus is defined as the ratio of shearing stress to shearing strain and measures a material's resistance to deformation via shear forces.
The document discusses Deepak's academic and professional background, including an MBA from IE Business School in Spain and experience founding perfectbazaar.com. It also provides an overview of the topics to be covered in the Strength of Materials course, such as stresses, strains, Hooke's law, and analysis of bars with varying cross-sections. The grading policy and syllabus are outlined which divide the course into 5 units covering various strength of materials concepts.
Dr. Salah Uddin teaches about relationships between elastic constants, including:
1. Young's modulus describes the proportionality between stress and strain in a simple tension or compression test.
2. Bulk modulus describes the ratio of direct stress to volumetric strain for mutually perpendicular, equal stresses.
3. Shear modulus describes the linear relationship between shear stress and shear strain.
4. Poisson's ratio describes the ratio of lateral to axial strain.
5. The elastic constants are related through equations involving Young's modulus, shear modulus, bulk modulus, and Poisson's ratio.
1.3 Stress & Strain Relationship of Hooke’s Law.pptVanathisekar2
Stress refers to external forces applied to a material, while strain refers to the deformation or change in shape of the material resulting from those stresses. Hooke's law states that within the elastic limit, the amount of strain produced is directly proportional to the stress applied. Different moduli describe the relationship between stress and strain, including Young's modulus, the bulk modulus, and the shear modulus. Stress and strain can be longitudinal, relating to changes in length, or transverse, relating to changes in width or thickness. The elastic limit is the maximum stress a material can withstand without permanent deformation, after which plastic deformation or fracture may occur.
This document provides an overview of fatigue failure resulting from variable loading. It discusses three main fatigue life methods: the stress-life method, strain-life method, and fracture mechanics method. The stress-life method is based on stress levels and is most commonly used for high-cycle fatigue predictions, while the strain-life method considers localized plastic deformation and is better for low-cycle applications. The fracture mechanics method assumes a crack is already present and is used to predict crack growth. Key concepts discussed include fatigue testing methods, S-N diagrams, endurance limits, stress concentrations, and cumulative damage.
Thermal stresses and strains occur when temperature changes cause materials to expand or contract. Thermal strain is proportional to the temperature change. Uneven heating can induce stresses if expansion is constrained. Thermal stresses are analyzed by considering both the thermal strain and any mechanical stresses from applied loads. Bars with non-uniform cross sections experience varying thermal stresses due to differences in expansion rates. Composite bars made of different materials also experience thermal stresses at their interface as the materials expand unequally with temperature changes. Thermal stresses can be calculated using equations that relate stress, strain, temperature change, elastic modulus, and coefficients of thermal expansion for the materials.
Mohr's circle is a graphical representation that illustrates the relationships between normal and shear stresses or strains at a point. It shows the two principal stresses or strains and the maximum shear stress or strain. The circle is centered at the average stress or strain and has a radius equal to the maximum shear value. Mohr's circle can be used to determine principal stresses/strains and directions, transform between stress/strain systems, and visualize how stresses/strains change with rotation. It remains a useful tool for engineers despite the availability of calculators.
This document summarizes the moment-area method for calculating deflections in beams. It discusses how the bending moment diagram can be divided into areas that correspond to rotations of the elastic curve. The sum of these areas multiplied by the distance to the centroid gives the tangential deviation, which can be used to determine the deflection. The method is applicable to statically indeterminate beams using superposition. Boundary conditions and how to handle different support types are also covered.
The document discusses plasticity theory and yield criteria. It introduces Hooke's law and its limitations under large strains. Generalized Hooke's law is presented for isotropic and anisotropic materials. Common stress-strain curves are shown including elastic-plastic and strain hardening responses. Several yield criteria are covered, including maximum principal stress, Tresca, and von Mises criteria. The von Mises criterion uses a second invariant of stress to predict yielding of ductile materials. An example compares predictions of yielding using Tresca and von Mises criteria for a given stress state in aluminum.
This document provides information about the Solid Mechanics course ME 302 taught by Dr. Nirmal Baran Hui at NIT Durgapur in West Bengal, India. It lists four required textbooks for the course and provides a detailed syllabus covering topics like stress, strain, elasticity, bending, deflection, columns, torsion, pressure vessels, combined loadings, springs, and failure theories. The document also includes examples of lecture content on stress analysis, stresses on oblique planes, and material subjected to pure shear.
Mohr circle (Complete Soil Mech. Undestanding Pakage: ABHAY)Abhay Kumar
This document discusses plane stress and strain transformations. It introduces concepts such as principal stresses/strains, maximum shear, and Mohr's circle. Mohr's circle can represent the state of stress or strain at a point on a plane. It allows determination of stresses/strains on any plane from knowledge of three independent quantities (e.g. sx, sy, txy). Strain gauges and rosettes are also discussed as tools to measure strain.
This document defines and describes different types of stresses and strains. Stress is the internal force per unit area that opposes deformation when a load is applied to a body. Strain is the deformation or change in length per original length of a body under an applied load. The types of stresses discussed include axial/direct stresses (tensile and compressive), shear stresses, bending stresses, combined stresses, and thermal stresses. Combined stress results from a combination of direct and indirect stresses, such as bending, torsional, and thermal stresses. Thermal stress is caused by changes in temperature and depends on the material's coefficient of thermal expansion.
1. The document discusses transformation equations that relate stresses on an inclined plane to stresses on a normal plane. The normal and shear stresses on the inclined plane can be determined by resolving the forces into components parallel and perpendicular to the inclined plane.
2. Principal stresses are the maximum and minimum normal stresses, which occur on principal planes where the shear stresses are zero. The principal stresses and planes can be determined by differentiating the transformation equations.
3. Maximum shear stress occurs at 45° to the normal plane and is equal to half the difference between the normal stresses.
1) Advanced metal forming techniques involve plastic deformation of metal crystals beyond their elastic limit.
2) Yield criteria determine which combination of multi-axial stresses will cause yielding, such as Tresca and von-Mises criteria relating to maximum shear stresses.
3) Mohr's circle provides a graphical representation of the state of stress at a point, showing normal and shear stresses on planes of all orientations. It can be used to determine principal stresses and maximum shear stress.
1) The document discusses Mohr's circle for analyzing stresses in plane stress, including how to construct Mohr's circle, determine principal stresses and maximum shear stresses, and stresses on inclined planes.
2) It provides the equations for transforming stresses between reference frames and deriving Mohr's circle.
3) An example is worked through applying Mohr's circle to determine stresses for a given stress element.
The document discusses compound stresses, which involve both normal and shear stresses acting on a plane. It provides equations to calculate:
1) Normal and shear stresses on a plane inclined to the given stress plane.
2) The inclination and normal stresses on the planes of maximum and minimum normal stress (principal planes).
3) The inclination and shear stresses on the planes of maximum shear stress.
It includes an example problem calculating the principal stresses and maximum shear stresses given a state of stress. Sign conventions for stresses are also defined.
Solution manual for introduction to finite element analysis and design nam ...Salehkhanovic
Solution Manual for Introduction to Finite Element Analysis and Design
Author(s) : Nam-Ho Kim and Bhavani V. Sankar
This solution manual include all problems (Chapters 0 to 8) of textbook.
This document discusses complex stress and strain resulting from combined loading. It begins by introducing plane stress and defining stress components acting on an infinitesimal element, including normal and shear stresses. It then derives equations to calculate stresses on inclined planes from the original stress components. These transformation equations allow determining principal stresses and angles, where normal stresses are maximum and minimum. Maximum shear stress can also be found and principal stresses calculated using properties of a right triangle. Principal angles correlate the two principal stresses. Shear stresses are zero on principal planes.
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 provides information on determining principal stresses and maximum shear stresses from given normal and shear stress values. It introduces Mohr's circle and the stress transformation equations, and shows how to calculate principal stresses and the angle of the principal stress planes using the equations. It also derives the equation to calculate maximum shear stress and shows the plane it acts on is 45 degrees from the principal planes. Several examples are worked through to demonstrate applying the equations.
This document provides equations and concepts related to the theory of elasticity and plasticity. It covers Cauchy stress formulas, stress components, principal stresses, strain tensors, constitutive equations including Hooke's law, boundary conditions, energy principles, and classical beam and bar theories. It also introduces the Ritz method for approximating displacements using basis functions to satisfy boundary conditions and minimize the total potential energy functional.
1) Mohr's circle is a graphical representation used to analyze stresses on planes at a point in a stressed body under plane stress conditions. It relates normal and shear stresses acting on inclined planes.
2) To construct Mohr's circle, the normal and shear stresses on the x and y faces of a sample are plotted on a stress diagram. A line is drawn between these points and extended to intersect the normal stress axis, establishing the center of the circle.
3) Mohr's circle allows visualization of relationships between stresses, including determining principal stresses, maximum shear stress, and planes on which they act from the circle's dimensions.
This book is intended to cover the basic Strength of Materials of the first
two years of an engineering degree or diploma course ; it does not attempt
to deal with the more specialized topics which usually comprise the final
year of such courses.
The work has been confined to the mathematical aspect of the subject
and no descriptive matter relating to design or materials testing has been
included.
This document discusses the analysis of laminated composite structures. It outlines the basic assumptions made in the analysis including linear strain-displacement and stress-strain relationships. It defines the strain-displacement relations and stress-strain relations for each layer of a laminate. Stress resultants and force-displacement relations are defined through laminate stiffness and compliance equations. Special classes of laminates are identified and the engineering properties of laminates are discussed. The analysis of laminated composite structures is then introduced.
Mohr's circle is a graphical representation of the transformation equations for plane stress. It allows visualization of normal and shear stresses on inclined planes at a point in a stressed body. Using Mohr's circle, one can calculate principal stresses, maximum shear stresses, and stresses on inclined planes. The procedure involves plotting the initial stress state (σx, τxy) and (σy, -τxy) and drawing a circle through these points. The principal and maximum shear stresses are then found by rotating the initial points. Mohr's circle provides a simple way to perform otherwise complicated stress analyses.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
Formula Bank and Important tips for Mechanical Engineering Students for Compe...Vinoth Jebaraj A
This document summarizes key concepts in engineering mechanics and strength of materials for mechanical engineering students. It covers topics like force equilibrium, stress and strain analysis, material properties, and failure theories. Key equations are presented for areas including static equilibrium, centroids, moments of inertia, stress-strain relationships, transformation of stresses, and bending stresses in beams. Diagrams illustrate stress distributions and Mohr's circle analyses for various loading conditions.
ESA Module 3 Part-B ME832. by Dr. Mohammed ImranMohammed Imran
This document discusses two-dimensional photoelasticity techniques for stress analysis. It describes various methods for separating principal stresses at interior points of a photoelastic model, including using a lateral extensometer, properties at free boundaries, Laplace's equation, shear-difference method, and oblique incidence method. It also covers scaling stresses between models and prototypes for various applications, including static, thermal, and dynamic cases. As an example, it discusses using photoelasticity to optimize the design of sheet pile sections for cofferdams.
Solution manual for introduction to nonlinear finite element analysis nam-h...Salehkhanovic
Solution Manual for Introduction to Nonlinear Finite Element Analysis
Author(s) : Nam-Ho Kim
This solution manual include all problems (Chapters 1 to 5) of textbook. There is one PDF for each of chapters.
This document discusses homogeneous linear systems with constant coefficients. It begins by defining such a system as x' = Ax, where A is an n×n matrix of real constants. It then explains that the equilibrium solutions are found by solving Ax = 0, and stability is determined by the eigenvalues of A. Examples are provided to illustrate finding the direction field, eigenvalues/eigenvectors, general solution, and phase plane plots for specific 2D systems. Time plots of the solutions are also shown.
This document provides an overview of geomechanics concepts for petroleum engineers. It discusses stress and strain theory, elasticity, homogeneous and heterogeneous stress fields, principal stresses, and the Mohr circle construction. It also covers rock deformation mechanisms including cataclasis and intracrystalline plasticity. Key concepts are defined such as normal and shear stress, elastic moduli like Young's modulus and Poisson's ratio, elastic stress-strain equations, and strain measures including conventional, quadratic, and natural strain.
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The document provides guidance on creating an effective CV and cover letter as well as tips for interview skills, including highlighting relevant qualifications and experience in reverse chronological order in the CV, customizing the cover letter for specific employers, and demonstrating strengths and how to handle challenging questions in interviews by providing real examples.
This document discusses stresses in pressure vessels and thin-walled approximations. It covers stresses in spherical and cylindrical pressure vessels. For thin-walled vessels with a ratio of inner radius to wall thickness over 10:1, the maximum normal stress is within 5% of the average normal stress. Cylindrical vessels experience hoop/tangential, axial/longitudinal, and radial stresses. Rotating rings experience similar tangential and radial stresses as thick-walled cylinders due to inertial forces. Press and shrink fits between cylinders create contact pressure at the interface.
This document reviews additive manufacturing research on cermets. Several powder-based AM processes have been used to produce cermets, including selective laser sintering/melting, laser engineering net shaping, direct laser deposition, and binder jet 3D printing. Research has focused on optimizing process parameters to produce dense, crack-free cermet components and evaluating properties like density and hardness. Key findings include that SLS/SLM of WC-Co is most widely studied and parameters like laser power, scan speed, layer thickness influence microstructure and properties of final parts.
This document discusses a study that evaluated the effect of indentation size on the microhardness values of two viscoelastic dental materials (a resin composite and resin modified glass ionomer cement) under different loads and holding times. Vickers and Knoop microhardness tests were used to assess the materials at loads of 100, 200, and 300g and times of 10, 20, and 30 seconds. The results showed that microhardness values for both materials were significantly affected by the load for Vickers hardness tests. For Knoop hardness tests, only the composite showed significant load-dependence while the resin modified glass ionomer cement did not. The optimal load and time to determine microhardness could not be conclusively determined for either
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This document describes Standard Test Method E 384 for determining microindentation hardness of materials. The standard covers using Knoop and Vickers indenters to make indentations between 1-1000 gf force. Hardness is calculated based on indentation size and geometry. Key points:
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The document provides guidance on who needs to self-isolate in New Brunswick and how to properly self-isolate. It states that individuals confirmed to have COVID-19, those with symptoms depending on their region's COVID-19 level, and all travelers returning to the province must self-isolate. When self-isolating, individuals must limit contact with others in the household, practice good hygiene, and isolate in separate living quarters if possible. If a separate bathroom cannot be maintained, the entire household must also self-isolate.
Colombo Dockyard PLC provides industrial training where trainees gain experience in various workshops like machinery outfitting, engine fitting, hull construction, and more. Trainees learn skills like pipe fitting, welding techniques, machining operations, and motor rewiring. The training also provides exposure to quality control processes, waste management techniques, and human resource practices at Colombo Dockyard. The training overall provided hands-on experience in ship repair activities and familiarized the trainee with engineering operations, maintenance, safety practices, and teamwork.
This study Examines the Effectiveness of Talent Procurement through the Imple...DharmaBanothu
In the world with high technology and fast
forward mindset recruiters are walking/showing interest
towards E-Recruitment. Present most of the HRs of
many companies are choosing E-Recruitment as the best
choice for recruitment. E-Recruitment is being done
through many online platforms like Linkedin, Naukri,
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Key Words : Talent Management, Talent Acquisition , E-
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Effectiveness of Talent Acquisition through E-
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AI in customer support Use cases solutions development and implementation.pdfmahaffeycheryld
AI in customer support will integrate with emerging technologies such as augmented reality (AR) and virtual reality (VR) to enhance service delivery. AR-enabled smart glasses or VR environments will provide immersive support experiences, allowing customers to visualize solutions, receive step-by-step guidance, and interact with virtual support agents in real-time. These technologies will bridge the gap between physical and digital experiences, offering innovative ways to resolve issues, demonstrate products, and deliver personalized training and support.
https://www.leewayhertz.com/ai-in-customer-support/#How-does-AI-work-in-customer-support
Sri Guru Hargobind Ji - Bandi Chor Guru.pdfBalvir Singh
Sri Guru Hargobind Ji (19 June 1595 - 3 March 1644) is revered as the Sixth Nanak.
• On 25 May 1606 Guru Arjan nominated his son Sri Hargobind Ji as his successor. Shortly
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Sikhs to be Saints and Soldier.
• He had a long tenure as Guru, lasting 37 years, 9 months and 3 days
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...PriyankaKilaniya
Energy efficiency has been important since the latter part of the last century. The main object of this survey is to determine the energy efficiency knowledge among consumers. Two separate districts in Bangladesh are selected to conduct the survey on households and showrooms about the energy and seller also. The survey uses the data to find some regression equations from which it is easy to predict energy efficiency knowledge. The data is analyzed and calculated based on five important criteria. The initial target was to find some factors that help predict a person's energy efficiency knowledge. From the survey, it is found that the energy efficiency awareness among the people of our country is very low. Relationships between household energy use behaviors are estimated using a unique dataset of about 40 households and 20 showrooms in Bangladesh's Chapainawabganj and Bagerhat districts. Knowledge of energy consumption and energy efficiency technology options is found to be associated with household use of energy conservation practices. Household characteristics also influence household energy use behavior. Younger household cohorts are more likely to adopt energy-efficient technologies and energy conservation practices and place primary importance on energy saving for environmental reasons. Education also influences attitudes toward energy conservation in Bangladesh. Low-education households indicate they primarily save electricity for the environment while high-education households indicate they are motivated by environmental concerns.
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We have designed & manufacture the Lubi Valves LBF series type of Butterfly Valves for General Utility Water applications as well as for HVAC applications.
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Digital Twins Computer Networking Paper Presentation.pptxaryanpankaj78
A Digital Twin in computer networking is a virtual representation of a physical network, used to simulate, analyze, and optimize network performance and reliability. It leverages real-time data to enhance network management, predict issues, and improve decision-making processes.
1. 1
Lecture 5
Plane Stress
Transformation Equations
Stress elements and plane stress.
Stresses on inclined sections.
Transformation equations.
Principal stresses, angles, and planes.
Maximum shear stress.
2. 2
Normal and shear stresses on inclined sections
To obtain a complete picture of the stresses in a bar, we must
consider the stresses acting on an “inclined” (as opposed to a
“normal”) section through the bar.
P
P
Normal section
Inclined section
Because the stresses are the same throughout the entire bar, the
stresses on the sections are uniformly distributed.
P Inclined
section
P Normal
section
3. 3
P
V
N
P
θ
x
y
The force P can be resolved into components:
Normal force N perpendicular to the inclined plane, N = P cos θ
Shear force V tangential to the inclined plane V = P sin θ
If we know the areas on which the forces act, we can calculate
the associated stresses.
x
y
area
A
area A
( / cos )
θ
σθ
x
y
τθ
area
A
area A
( / cos )
θ
5. 5
Introduction to stress elements
Stress elements are a useful way to represent stresses acting at some
point on a body. Isolate a small element and show stresses acting on all
faces. Dimensions are “infinitesimal”, but are drawn to a large scale.
x
y
z
σ =
x P A
/
σx
x
y
σx
σ =
x P A
/
P σx = A
P /
x
y
Area A
6. 6
Maximum stresses on a bar in tension
P
P
a
σx = σmax = P / A
σx
a
Maximum normal stress,
Zero shear stress
7. 7
P
P
a
b
Maximum stresses on a bar in tension
b
σx/2
σx/2
τmax = σx/2
θ = 45°
Maximum shear stress,
Non-zero normal stress
8. 8
Stress Elements and Plane Stress
When working with stress elements, keep in mind that only
one intrinsic state of stress exists at a point in a stressed
body, regardless of the orientation of the element used to
portray the state of stress.
We are really just rotating axes to represent stresses in a
new coordinate system.
x
y
σx σx
θ
y1
x1
9. 9
y
z
x
Normal stresses σx, σy, σz
(tension is positive)
σx
σx
σy
σy
σz
τxy
τyx
τxz
τzx
τzy
τyz
Shear stresses τxy = τyx,
τxz = τzx, τyz = τzy
Sign convention for τab
Subscript a indicates the “face” on which the stress acts
(positive x “face” is perpendicular to the positive x direction)
Subscript b indicates the direction in which the stress acts
Strictly σx = σxx, σy = σyy, σz = σzz
10. 10
When an element is in plane stress in the xy plane, only the x and
y faces are subjected to stresses (σz = 0 and τzx = τxz = τzy = τyz = 0).
Such an element could be located on the free surface of a body (no
stresses acting on the free surface).
σx σx
σy
σy
τxy
τyx
τxy
τyx
x
y
Plane stress element in 2D
σx, σy
τxy = τyx
σz = 0
11. 11
Stresses on Inclined Sections
σx σx
σy
τxy
τyx
τxy
τyx
x
y
x
y
σx1
θ
y1
x1
σx1
σy1
σy1
τx y
1 1
τy x
1 1
τx y
1 1
τy x
1 1
The stress system is known in terms of coordinate system xy.
We want to find the stresses in terms of the rotated coordinate
system x1y1.
Why? A material may yield or fail at the maximum value of σ or τ. This
value may occur at some angle other than θ = 0. (Remember that for uni-
axial tension the maximum shear stress occurred when θ = 45 degrees. )
12. 12
Transformation Equations
y
σx1
x
θ
y1
x1
σx
σy
τx y
1 1
τxy
τyx
θ
Stresses
y
σ θ
x1 A sec
x
θ
y1
x1
σxA
σ θ
y A tan
τ θ
x y
1 1 A sec
τxy A
τ θ
yx A tan
θ
Forces
Left face has area A.
Bottom face has area A tan θ.
Inclined face has area A sec θ.
Forces can be found from stresses if
the area on which the stresses act is
known. Force components can then
be summed.
13. 13
y
σ θ
x1 A sec
x
θ
y1
x1
σxA
σ θ
y A tan
τ θ
x y
1 1 A sec
τxy A
τ θ
yx A tan
θ
( ) ( ) ( ) ( ) 0
cos
tan
sin
tan
sin
cos
sec
:
direction
in the
forces
Sum
1
1
=
−
−
−
− θ
θ
τ
θ
θ
σ
θ
τ
θ
σ
θ A
A
A
A
A
σ
x
yx
y
xy
x
x
( ) ( ) ( ) ( ) 0
sin
tan
cos
tan
cos
sin
sec
:
direction
in the
forces
Sum
1
1
1
=
−
−
−
+ θ
θ
τ
θ
θ
σ
θ
τ
θ
σ
θ
τ A
A
A
A
A
y
yx
y
xy
x
y
x
( ) ( )
θ
θ
τ
θ
θ
σ
σ
τ
θ
θ
τ
θ
σ
θ
σ
σ
τ
τ
2
2
1
1
2
2
1
sin
cos
cos
sin
cos
sin
2
sin
cos
:
gives
g
simplifyin
and
Using
−
+
−
−
=
+
+
=
=
xy
y
x
y
x
xy
y
x
x
yx
xy
15. 15
Example: The state of plane stress at a point is represented by the
stress element below. Determine the stresses acting on an element
oriented 30° clockwise with respect to the original element.
80 MPa 80 MPa
50 MPa
x
y
50 MPa
25 MPa
Define the stresses in terms of the
established sign convention:
σx = -80 MPa σy = 50 MPa
τxy = -25 MPa
We need to find σx1, σy1, and
τx1y1 when θ = -30°.
Substitute numerical values into the transformation equations:
( ) ( ) ( ) MPa
9
.
25
30
2
sin
25
30
2
cos
2
50
80
2
50
80
2
sin
2
cos
2
2
1
1
−
=
°
−
−
+
°
−
−
−
+
+
−
=
+
−
+
+
=
x
xy
y
x
y
x
x
σ
θ
τ
θ
σ
σ
σ
σ
σ
16. 16
( )
( ) ( ) ( ) ( ) MPa
8
.
68
30
2
cos
25
30
2
sin
2
50
80
2
cos
2
sin
2
1
1
1
1
−
=
°
−
−
+
°
−
−
−
−
=
+
−
−
=
y
x
xy
y
x
y
x
τ
θ
τ
θ
σ
σ
τ
( ) ( ) ( ) MPa
15
.
4
30
2
sin
25
30
2
cos
2
50
80
2
50
80
2
sin
2
cos
2
2
1
1
−
=
°
−
−
−
°
−
−
−
−
+
−
=
−
−
−
+
=
y
xy
y
x
y
x
y
σ
θ
τ
θ
σ
σ
σ
σ
σ
Note that σy1 could also be obtained
(a) by substituting +60° into the
equation for σx1 or (b) by using the
equation σx + σy = σx1 + σy1
+60°
25.8 MPa
25.8 MPa
4.15 MPa
x
y
4.15 MPa
68.8 MPa
x1
y1
-30
o
(from Hibbeler, Ex. 15.2)
17. 17
Principal Stresses
The maximum and minimum normal stresses (σ1 and σ2) are
known as the principal stresses. To find the principal stresses,
we must differentiate the transformation equations.
( ) ( )
( )
y
x
xy
p
xy
y
x
x
xy
y
x
x
xy
y
x
y
x
x
d
d
d
d
σ
σ
τ
θ
θ
τ
θ
σ
σ
θ
σ
θ
τ
θ
σ
σ
θ
σ
θ
τ
θ
σ
σ
σ
σ
σ
−
=
=
+
−
−
=
=
+
−
−
=
+
−
+
+
=
2
2
tan
0
2
cos
2
2
sin
0
2
cos
2
2
sin
2
2
2
sin
2
cos
2
2
1
1
1
θp are principal angles associated with
the principal stresses
There are two values of 2θp in the range 0-360°, with values differing by 180°.
There are two values of θp in the range 0-180°, with values differing by 90°.
So, the planes on which the principal stresses act are mutually perpendicular.
18. 18
θ
τ
θ
σ
σ
σ
σ
σ
σ
σ
τ
θ
2
sin
2
cos
2
2
2
2
tan
1 xy
y
x
y
x
x
y
x
xy
p
+
−
+
+
=
−
=
We can now solve for the principal stresses by substituting for θp
in the stress transformation equation for σx1. This tells us which
principal stress is associated with which principal angle.
2θp
τxy
(σx – σy) / 2
R
R
R
R
xy
p
y
x
p
xy
y
x
τ
θ
σ
σ
θ
τ
σ
σ
=
−
=
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
=
2
sin
2
2
cos
2
2
2
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
−
+
+
=
R
R
xy
xy
y
x
y
x
y
x τ
τ
σ
σ
σ
σ
σ
σ
σ
2
2
2
1
19. 19
Substituting for R and re-arranging gives the larger of the two
principal stresses:
2
2
1
2
2
xy
y
x
y
x
τ
σ
σ
σ
σ
σ +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
+
+
=
To find the smaller principal stress, use σ1 + σ2 = σx + σy.
2
2
1
2
2
2
xy
y
x
y
x
y
x τ
σ
σ
σ
σ
σ
σ
σ
σ +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
−
+
=
−
+
=
These equations can be combined to give:
2
2
2
,
1
2
2
xy
y
x
y
x
τ
σ
σ
σ
σ
σ +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
±
+
=
Principal stresses
To find out which principal stress goes with which principal angle,
we could use the equations for sin θp and cos θp or for σx1.
20. 20
The planes on which the principal stresses act are called the
principal planes. What shear stresses act on the principal planes?
Solving either equation gives the same expression for tan 2θp
Hence, the shear stresses are zero on the principal planes.
( )
( )
( ) 0
2
cos
2
2
sin
0
2
cos
2
2
sin
0
2
cos
2
sin
2
0
and
0
for
equations
the
Compare
1
1
1
1
1
1
=
+
−
−
=
=
+
−
−
=
+
−
−
=
=
=
θ
τ
θ
σ
σ
θ
σ
θ
τ
θ
σ
σ
θ
τ
θ
σ
σ
τ
θ
σ
τ
xy
y
x
x
xy
y
x
xy
y
x
y
x
x
y
x
d
d
d
d
24. 24
The two principal stresses determined so far are the principal
stresses in the xy plane. But … remember that the stress element
is 3D, so there are always three principal stresses.
y
x
σx
σx
σy
σy
τ
τ
τ
τ
σx, σy, τxy = τyx = τ
yp
zp
xp
σ1
σ1
σ2
σ2
σ3 = 0
σ1, σ2, σ3 = 0
Usually we take σ1 > σ2 > σ3. Since principal stresses can be com-
pressive as well as tensile, σ3 could be a negative (compressive)
stress, rather than the zero stress.
25. 25
Maximum Shear Stress
( )
( )
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
−
=
=
−
−
−
=
+
−
−
=
xy
y
x
s
xy
y
x
y
x
xy
y
x
y
x
d
d
τ
σ
σ
θ
θ
τ
θ
σ
σ
θ
τ
θ
τ
θ
σ
σ
τ
2
2
tan
0
2
sin
2
2
cos
2
cos
2
sin
2
1
1
1
1
To find the maximum shear stress, we must differentiate the trans-
formation equation for shear.
There are two values of 2θs in the range 0-360°, with values differing by 180°.
There are two values of θs in the range 0-180°, with values differing by 90°.
So, the planes on which the maximum shear stresses act are mutually
perpendicular.
Because shear stresses on perpendicular planes have equal magnitudes,
the maximum positive and negative shear stresses differ only in sign.
26. 26
2θs
τxy
(σ
x
–
σ
y
)
/
2
R
( ) θ
τ
θ
σ
σ
τ
τ
σ
σ
θ
2
cos
2
sin
2
2
2
tan
1
1 xy
y
x
y
x
xy
y
x
s
+
−
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
−
=
We can now solve for the maximum shear stress by substituting for
θs in the stress transformation equation for τx1y1.
R
R
R
y
x
s
xy
s
xy
y
x
2
2
sin
2
cos
2
2
2
2
σ
σ
θ
τ
θ
τ
σ
σ
−
−
=
=
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
=
max
min
2
2
max
2
τ
τ
τ
σ
σ
τ −
=
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
= xy
y
x
27. 27
Use equations for sin θs and cos θs or τx1y1 to find out which face
has the positive shear stress and which the negative.
What normal stresses act on the planes with maximum shear stress?
Substitute for θs in the equations for σx1 and σy1 to get
s
y
x
y
x σ
σ
σ
σ
σ =
+
=
=
2
1
1
x
y
σs
θs
σs
σs
σs
τmax
τmax
τmax
τmax
σy
σx σx
σy
τxy
τyx
τxy
τyx
x
y
28. 28
Example: The state of plane stress at a point is represented by the
stress element below. Determine the maximum shear stresses and
draw the corresponding stress element.
80 MPa 80 MPa
50 MPa
x
y
50 MPa
25 MPa
Define the stresses in terms of the
established sign convention:
σx = -80 MPa σy = 50 MPa
τxy = -25 MPa
( ) MPa
6
.
69
25
2
50
80
2
2
2
max
2
2
max
=
−
+
⎟
⎠
⎞
⎜
⎝
⎛ −
−
=
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
=
τ
τ
σ
σ
τ xy
y
x
MPa
15
2
50
80
2
−
=
+
−
=
+
=
s
y
x
s
σ
σ
σ
σ
30. 30
Finally, we can ask how the principal stresses and maximum shear
stresses are related and how the principal angles and maximum
shear angles are related.
2
2
2
2
2
2
2
1
max
max
2
1
2
2
2
1
2
2
2
,
1
σ
σ
τ
τ
σ
σ
τ
σ
σ
σ
σ
τ
σ
σ
σ
σ
σ
−
=
=
−
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
=
−
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
±
+
=
xy
y
x
xy
y
x
y
x
p
p
s
y
x
xy
p
xy
y
x
s
θ
θ
θ
σ
σ
τ
θ
τ
σ
σ
θ
2
cot
2
tan
1
2
tan
2
2
tan
2
2
tan
−
=
−
=
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
−
=
32. 32
80 MPa 80 MPa
50 MPa
x
y
50 MPa
25 MPa
Original Problem
σx = -80, σy = 50, τxy = 25
84.6 MPa
84.6 MPa
54.6 MPa
x
y
54.6 MPa
10.5
o
100.5
o
Principal Stresses
σ1 = 54.6, σ2 = 0, σ3 = -84.6
15 MPa
15 MPa
15 MPa
x
y
15 MPa
69.6 MPa
-34.5o
55.5o
Maximum Shear
τmax = 69.6, σs = -15
°
°
−
=
°
±
°
=
°
±
=
5
.
55
,
5
.
34
45
5
.
10
45
s
s
p
s
θ
θ
θ
θ
( )
MPa
6
.
69
2
6
.
84
6
.
54
2
max
max
2
1
max
=
−
−
=
−
=
τ
τ
σ
σ
τ
33. 1
Mohr’s Circle for Plane Stress
Transformation equations for plane stress.
Procedure for constructing Mohr’s circle.
Stresses on an inclined element.
Principal stresses and maximum shear stresses.
Introduction to the stress tensor.
34. 2
Stress Transformation Equations
σx σx
σy
τxy
τyx
τxy
τyx
x
y
x
y
σx1
θ
y1
x1
σx1
σy1
σy1
τx y
1 1
τy x
1 1
τx y
1 1
τy x
1 1
( ) θ
τ
θ
σ
σ
τ
θ
τ
θ
σ
σ
σ
σ
σ
2
cos
2
sin
2
2
sin
2
cos
2
2
1
1
1
xy
y
x
y
x
xy
y
x
y
x
x
+
−
−
=
+
−
+
+
=
If we vary θ from 0° to 360°, we will get all possible values of σx1 and τx1y1
for a given stress state. It would be useful to represent σx1 and τx1y1 as
functions of θ in graphical form.
35. 3
To do this, we must re-write the transformation equations.
( ) θ
τ
θ
σ
σ
τ
θ
τ
θ
σ
σ
σ
σ
σ
2
cos
2
sin
2
2
sin
2
cos
2
2
1
1
1
xy
y
x
y
x
xy
y
x
y
x
x
+
−
−
=
+
−
=
+
−
Eliminate θ by squaring both sides of each equation and adding
the two equations together.
2
2
2
1
1
2
1
2
2
xy
y
x
y
x
y
x
x τ
σ
σ
τ
σ
σ
σ +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
=
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ +
−
Define σavg and R
2
2
2
2
xy
y
x
y
x
avg R τ
σ
σ
σ
σ
σ +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
=
+
=
36. 4
Substitue for σavg and R to get
( ) 2
2
1
1
2
1 R
y
x
avg
x =
+
− τ
σ
σ
which is the equation for a circle with centre (σavg,0) and radius R.
This circle is usually referred to as
Mohr’s circle, after the German civil
engineer Otto Mohr (1835-1918). He
developed the graphical technique for
drawing the circle in 1882.
The construction of Mohr’s circle is
one of the few graphical techniques
still used in engineering. It provides
a simple and clear picture of an
otherwise complicated analysis.
37. 5
Sign Convention for Mohr’s Circle
x
y
σx1
θ
y1
x1
σx1
σy1
σy1
τx y
1 1
τy x
1 1
τx y
1 1
τy x
1 1
( ) 2
2
1
1
2
1 R
y
x
avg
x =
+
− τ
σ
σ
σx1
τx1y1
R
2θ
σavg
Notice that shear stress is plotted as positive downward.
The reason for doing this is that 2θ is then positive counterclockwise,
which agrees with the direction of 2θ used in the derivation of the
tranformation equations and the direction of θ on the stress element.
Notice that although 2θ appears in Mohr’s circle, θ appears on the
stress element.
38. 6
Procedure for Constructing Mohr’s Circle
1. Draw a set of coordinate axes with σx1 as abscissa (positive to the
right) and τx1y1 as ordinate (positive downward).
2. Locate the centre of the circle c at the point having coordinates σx1
= σavg and τx1y1 = 0.
3. Locate point A, representing the stress conditions on the x face of
the element by plotting its coordinates σx1 = σx and τx1y1 = τxy. Note
that point A on the circle corresponds to θ = 0°.
4. Locate point B, representing the stress conditions on the y face of
the element by plotting its coordinates σx1 = σy and τx1y1 = −τxy.
Note that point B on the circle corresponds to θ = 90°.
5. Draw a line from point A to point B, a diameter of the circle passing
through point c. Points A and B (representing stresses on planes
at 90° to each other) are at opposite ends of the diameter (and
therefore 180° apart on the circle).
6. Using point c as the centre, draw Mohr’s circle through points A
and B. This circle has radius R.
(based on Gere)
40. 8
Stresses on an Inclined Element
1. On Mohr’s circle, measure an angle 2θ counterclockwise from
radius cA, because point A corresponds to θ = 0 and hence is
the reference point from which angles are measured.
2. The angle 2θ locates the point D on the circle, which has
coordinates σx1 and τx1y1. Point D represents the stresses on the
x1 face of the inclined element.
3. Point E, which is diametrically opposite point D on the circle, is
located at an angle 2θ + 180° from cA (and 180° from cD). Thus
point E gives the stress on the y1 face of the inclined element.
4. So, as we rotate the x1y1 axes counterclockwise by an angle θ,
the point on Mohr’s circle corresponding to the x1 face moves
counterclockwise through an angle 2θ.
(based on Gere)
43. 11
Maximum Shear Stress
σx σx
σy
τxy
τyx
τxy
τyx
x
y
A (θ=0)
σx1
τx1y1
c
R
B (θ=90)
A
B
2θs
σs
τmax
τmin
x
y
σs
θs
σs
σs
σs
τmax
τmax
τmax
τmax
Note carefully the
directions of the
shear forces.
44. 12
Example: The state of plane stress at a point is represented by the stress
element below. Draw the Mohr’s circle, determine the principal stresses and
the maximum shear stresses, and draw the corresponding stress elements.
80 MPa 80 MPa
50 MPa
x
y
50 MPa
25 MPa
σ
τ
15
2
50
80
2
−
=
+
−
=
+
=
=
y
x
avg
c
σ
σ
σ
c
A (θ=0)
A
B (θ=90)
B
( )
( ) ( )
6
.
69
25
65
25
15
50
2
2
2
2
=
+
=
+
−
−
=
R
R
R
σ1
σ2
MPa
6
.
84
MPa
6
.
54
6
.
69
15
2
1
2
,
1
2
,
1
−
=
=
±
−
=
±
=
σ
σ
σ
σ R
c
τmax
MPa
15
MPa
6
.
69
s
max
−
=
=
=
=
c
R
σ
τ
45. 13
84.6 MPa
84.6 MPa
54.6 MPa
x
y
54.6 MPa
10.5o
100.5
o
σ
τ
c
A (θ=0)
B (θ=90)
R
80 MPa 80 MPa
50 MPa
x
y
50 MPa
25 MPa
σ1
σ2
2θ2
2θ1
°
=
°
=
°
=
°
+
=
°
=
=
−
=
5
.
10
5
.
100
201
180
0
.
21
2
0
.
21
2
3846
.
0
15
80
25
2
tan
2
1
1
2
2
θ
θ
θ
θ
θ
2θ
46. 14
σ
τ
c
A (θ=0)
B (θ=90)
R
80 MPa 80 MPa
50 MPa
x
y
50 MPa
25 MPa
2θ2
2θ
15 MPa
15 MPa
15 MPa
x
y
15 MPa
69.6 MPa
-34.5o
55.5
o
2θsmax
°
=
°
=
°
+
=
°
=
5
.
55
0
.
111
90
0
.
21
2
0
.
21
2
max
max
2
s
s
θ
θ
θ
2θsmin
taking sign convention into
account
°
−
=
°
−
=
−
−
=
°
=
5
.
34
0
.
69
)
0
.
21
90
(
2
0
.
21
2
min
min
2
s
s
θ
θ
θ
τmax
τmin
47. 15
80 MPa 80 MPa
50 MPa
x
y
50 MPa
25 MPa
A (θ=0)
σ
τ
B (θ=90)
25.8 MPa
25.8 MPa
4.15 MPa
x
y
4.15 MPa
68.8 MPa
x1
y1
-30
o
Example: The state of plane stress at a point is represented by the stress
element below. Find the stresses on an element inclined at 30° clockwise
and draw the corresponding stress elements.
-60°
-60+180°
C (θ = -30°)
C
D (θ = -30+90°)
D
2θ2
σx1 = c – R cos(2θ2+60)
σy1 = c + R cos(2θ2+60)
τx1y1= -R sin (2θ2+60)
σx1 = -26
σy1 = -4
τx1y1= -69
2θ
θ = -30°
2θ = -60°
48. 16
σ
τ
A (θ=0)
B (θ=90)
Principal Stresses σ1 = 54.6 MPa, σ2 = -84.6 MPa
But we have forgotten about the third principal stress!
Since the element is in plane stress (σz = 0),
the third principal stress is zero.
σ1 = 54.6 MPa
σ2 = 0 MPa
σ3 = -84.6 MPa
σ1
σ2
σ3
This means three
Mohr’s circles can
be drawn, each
based on two
principal stresses:
σ1 and σ3
σ1 and σ2
σ2 and σ3
52. 20
From our analyses so far, we know that for a given stress system,
it is possible to find a set of three principal stresses. We also know
that if the principal stresses are acting, the shear stresses must be
zero. In terms of the stress tensor,
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
zz
zy
zx
yz
yy
yx
xz
xy
xx
σ
τ
τ
τ
σ
τ
τ
τ
σ
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
3
2
1
0
0
0
0
0
0
σ
σ
σ
In mathematical terms, this is the process of matrix diagonaliza-
tion in which the eigenvalues of the original matrix are just the
principal stresses.
53. 21
80 MPa 80 MPa
50 MPa
x
y
50 MPa
25 MPa
Example: The state of plane stress at a point is represented by the
stress element below. Find the principal stresses.
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
50
25
25
80
y
yx
xy
x
M
σ
τ
τ
σ
We must find the eigenvalues of
this matrix.
Remember the general idea of eigenvalues. We are looking
for values of λ such that:
Ar = λr where r is a vector, and A is a matrix.
Ar – λr = 0 or (A – λI) r = 0 where I is the identity matrix.
For this equation to be true, either r = 0 or det (A – λI) = 0.
Solving the latter equation (the “characteristic equation”)
gives us the eigenvalues λ1 and λ2.
54. 22
6
.
54
,
6
.
84
0
4625
30
0
)
25
)(
25
(
)
50
)(
80
(
0
50
25
25
80
det
2
−
=
=
−
+
=
−
−
−
−
−
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
−
−
−
λ
λ
λ
λ
λ
λ
λ
So, the principal stresses are –84.6 MPa and
54.6 MPa, as before.
Knowing the eigenvalues, we can find the eigenvectors. These can be
used to find the angles at which the principal stresses act. To find the
eigenvectors, we substitute the eigenvalues into the equation (A – λI ) r
= 0 one at a time and solve for r.
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
−
−
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
−
−
−
0
0
6
.
54
50
25
25
6
.
54
80
0
0
50
25
25
80
y
x
y
x
λ
λ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
−
−
1
186
.
0
186
.
0
0
0
64
.
4
25
25
6
.
134
y
x
y
x
is one eigenvector.