This document discusses stress-strain relationships in materials subjected to axial loads. It covers key concepts such as elastic and plastic deformation, ductile and brittle behavior, stress-strain diagrams, and the effects of temperature, strain rate, and time-dependent behavior like creep and stress relaxation. Measurement techniques for strain like strain gages and extensometers are also described. Various stress-strain models are presented, including Hooke's law, the Ramberg-Osgood equation, and idealized perfectly plastic, elastic-plastic, and strain hardening models. The relationships between stress, strain, elastic modulus, yield strength, and other mechanical properties are examined through diagrams and equations.
Strength of Materials Lecture - 2
Elastic stress and strain of materials (stress-strain diagram)
Mehran University of Engineering and Technology.
Department of Mechanical Engineering.
Ekeeda is an online portal which creates and provides exclusive content for all branches engineering.To have more updates you can goto www.ekeeda.com..or you can contact on 8433429809...
1. The document discusses principal stresses and planes, describing how to determine the maximum and minimum normal stresses (principal stresses) and their corresponding planes from a state of plane stress.
2. It introduces Mohr's circle as a graphical method to determine principal stresses and maximum shear stresses from the stresses on any plane.
3. Equations are derived relating the principal stresses and maximum shear stress to the normal and shear stresses on any plane using trigonometric functions of the angle between the plane and principal planes.
This document provides an introduction and overview of a course on the theory of elasticity and plasticity. It discusses key topics that will be covered, including stress, strain, material behavior, elasticity problem formulation, energy principles, finite element methods, and plasticity. It outlines the organization of the course, including lecture materials made available online, instructors, office hours, grading based on homework, exams, and participation. Mathematical preliminaries are also introduced, including definitions and properties of scalars, vectors, tensors, and transformations between coordinate systems.
This document gives the class notes of Unit 2 stresses in composite sections. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
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.
Strength of Materials Lecture - 2
Elastic stress and strain of materials (stress-strain diagram)
Mehran University of Engineering and Technology.
Department of Mechanical Engineering.
Ekeeda is an online portal which creates and provides exclusive content for all branches engineering.To have more updates you can goto www.ekeeda.com..or you can contact on 8433429809...
1. The document discusses principal stresses and planes, describing how to determine the maximum and minimum normal stresses (principal stresses) and their corresponding planes from a state of plane stress.
2. It introduces Mohr's circle as a graphical method to determine principal stresses and maximum shear stresses from the stresses on any plane.
3. Equations are derived relating the principal stresses and maximum shear stress to the normal and shear stresses on any plane using trigonometric functions of the angle between the plane and principal planes.
This document provides an introduction and overview of a course on the theory of elasticity and plasticity. It discusses key topics that will be covered, including stress, strain, material behavior, elasticity problem formulation, energy principles, finite element methods, and plasticity. It outlines the organization of the course, including lecture materials made available online, instructors, office hours, grading based on homework, exams, and participation. Mathematical preliminaries are also introduced, including definitions and properties of scalars, vectors, tensors, and transformations between coordinate systems.
This document gives the class notes of Unit 2 stresses in composite sections. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
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.
The document discusses key concepts related to elastic, homogeneous, and isotropic materials including: limits of proportionality and elasticity, yield limit, ultimate strength, strain hardening, proof stress, and the stress-strain relationships of ductile and brittle materials. It provides definitions and examples for each term and describes the stress-strain graphs for ductile materials like mild steel and brittle materials.
The document discusses different types of loads acting on columns and how they induce stresses. It defines axial load, eccentric load, and eccentricity. It explains that eccentric loads produce both direct and bending stresses while axial loads only produce direct stresses. It provides equations to calculate the maximum and minimum stresses in a column under eccentric loading. An example problem is worked out calculating the maximum stresses in a T-section column loaded eccentrically. The document also discusses loads, eccentricity, and stresses on dams, retaining walls, and chimneys/walls loaded by wind pressure.
The document discusses buckling and its theories. It defines buckling as the failure of a slender structural member subjected to compressive loads. It provides examples of structures that can experience buckling. It explains Euler's theory of buckling which derived an equation for the critical buckling load of a long column based on its bending stress. The assumptions of Euler's theory are listed. Four cases of long column buckling based on end conditions are examined: both ends pinned, both ends fixed, one end fixed and one end pinned, one end fixed and one end free. Effective lengths are defined for each case and the corresponding critical buckling loads given. Limitations of Euler's theory are noted. Rankine's empirical formula for calculating ultimate
This document provides information about a Mechanics of Materials course, including:
- The instructor's name and credentials.
- An overview of course contents including stresses, strains, torsion, bending, centroids, and beam deflection methods.
- An introduction to mechanics of materials and the objectives to analyze stresses, strains, and displacements in structures.
- Key terms like stress, strain, axial force, normal force, shear force, deformation, prismatic and non-prismatic bars are defined.
- The stress-strain diagram is discussed and key points like the elastic region, proportional limit, yield point, strain hardening, ultimate stress, and necking are explained.
This document discusses stresses in beams and beam deflection. It covers several methods for analyzing bending stresses and deflection in beams, including: [1] the engineering beam theory relating moment, curvature, and stress; [2] double integration and moment area methods for calculating slope and deflection; and [3] Macaulay's method, which simplifies calculations for beams with eccentric loads. Formulas are provided relating bending moment, shear force, curvature, slope, and deflection. Moment-area theorems are also described for relating bending moment to slope and deflection.
Mohr's circle is a graphical representation of the transformation of stresses on planes at a point in a material. It relates normal and shear stresses on inclined planes to the principal stresses. The circle is centered at the average stress and has a radius equal to the difference between the maximum and minimum principal stresses. Mohr's circle allows determination of stresses on any inclined plane from knowledge of the principal stresses and provides insight into failure conditions of materials.
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.
Objective of the experiment:
1 - Study the relationship between the force (P) and
elongation (ΔL).
2 - Stability and study the relationship between strain (ε)
and stress (σ).
3 - Study the concept of the mechanical properties of solids.
4 - Establish a modulus of elasticity (E)
The document discusses stress and strain in engineering structures. It defines load, stress, strain and different types of each. Stress is the internal resisting force per unit area within a loaded component. Strain is the ratio of dimensional change to original dimension of a loaded body. Loads can be tensile, compressive or shear. Hooke's law states stress is proportional to strain within the elastic limit. The elastic modulus defines this proportionality. A tensile test measures the stress-strain curve, identifying elastic limit and other failure points. Multi-axial stress-strain relationships follow Poisson's ratio definitions.
The document discusses different types of strain energy stored in materials when subjected to loads. It defines strain energy as the work done or energy stored in a body during elastic deformation. The types of strain energy discussed include: elastic strain energy, strain energy due to gradual, sudden, impact, shock and shear loading. Formulas are provided to calculate strain energy due to these different loadings. Examples of calculating strain energy in axially loaded bars and beams subjected to bending and torsional loads are also presented.
Structural Mechanics: Shear stress in Beams (1st-Year)Alessandro Palmeri
- The document discusses shear stress in beams, specifically focusing on Jourawski's formula for calculating shear stress.
- Jourawski's formula provides an approximate solution for the shear stress distribution over a beam cross-section using simple equilibrium considerations.
- The formula can be applied to solid rectangular sections, giving a parabolic shear stress distribution, but provides inaccurate results for T-section and I-section flanges.
- For T-sections and I-sections, the formula can be applied to the web, where it accurately models the shear stress as parabolic. The maximum shear stress occurs at the neutral axis.
This document summarizes concepts related to torsion and the torsion of circular elastic bars. It discusses the assumptions made in analyzing torsion, including that shear strain varies linearly from the central axis. It also covers determining shear stress and torque using the polar moment of inertia for circular cross-sections. The relationships between applied torque, shear stress, shear strain, and angle of twist are defined. Stress concentrations and alternative differential equations approaches are also summarized.
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.
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.
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.
Maximum principal stress theory.
Maximum shear stress theory.
Maximum shear strain theory.
Maximum strain energy theory.
Maximum shear strain energy theory.
This document gives the class notes of Unit 6: Bending and shear Stresses in beams. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
- Stress is defined as force per unit area and can be divided into normal and shear components at a point. Stress around a point in 3D forms a stress ellipsoid with three orthogonal principal stress directions.
- Strain is the change in size and shape of a body due to applied stresses. It includes extension, shear and changes to the ellipsoid shape defined by finite stretches.
- The relationship between stress and strain is evaluated through rock deformation experiments using triaxial apparatus to measure shortening, strain rates, and ductility. The results relate to the rheology and deformation mechanisms in rocks.
This document provides an overview of topics related to strength of materials and mechanics of solids, including normal stress and strain, shear stress and strain, strain energy, impact loads, principal stress and strain, Mohr's stress circle, equilibrium equations, Hooke's law, and theories of failure. It includes definitions, formulas, and examples for each topic.
1. The document discusses unsymmetrical bending of beams. When a beam bends about an axis that is not perpendicular to a plane of symmetry, it is undergoing unsymmetrical bending.
2. Key aspects discussed include determining the principal axes, direct stress distribution, and deflection of beams under unsymmetrical bending. Equations are provided to calculate stresses and deflections.
3. An example problem is given involving finding the stresses at two points on a cantilever beam subjected to an unsymmetrical loading. The principal moments of inertia and neutral axis orientation are calculated.
This document discusses mechanical properties and testing methods. It introduces key terms like stress, strain, tensile testing and how properties like Young's modulus, yield strength and toughness are obtained. Tensile testing provides a stress-strain curve that shows elastic and plastic deformation regions. Ceramics are more brittle so bend testing is used to determine properties like flexural strength. Hardness tests measure a material's resistance to penetration.
Terminology for Mechanical Properties The Tensile Test: Stress-Strain Diagram...manohar3970
Terminology for Mechanical Properties
The Tensile Test: Stress-Strain Diagram
Properties Obtained from a Tensile Test
True Stress and True Strain
The Bend Test for Brittle Materials
Hardness of Materials
The document discusses key concepts related to elastic, homogeneous, and isotropic materials including: limits of proportionality and elasticity, yield limit, ultimate strength, strain hardening, proof stress, and the stress-strain relationships of ductile and brittle materials. It provides definitions and examples for each term and describes the stress-strain graphs for ductile materials like mild steel and brittle materials.
The document discusses different types of loads acting on columns and how they induce stresses. It defines axial load, eccentric load, and eccentricity. It explains that eccentric loads produce both direct and bending stresses while axial loads only produce direct stresses. It provides equations to calculate the maximum and minimum stresses in a column under eccentric loading. An example problem is worked out calculating the maximum stresses in a T-section column loaded eccentrically. The document also discusses loads, eccentricity, and stresses on dams, retaining walls, and chimneys/walls loaded by wind pressure.
The document discusses buckling and its theories. It defines buckling as the failure of a slender structural member subjected to compressive loads. It provides examples of structures that can experience buckling. It explains Euler's theory of buckling which derived an equation for the critical buckling load of a long column based on its bending stress. The assumptions of Euler's theory are listed. Four cases of long column buckling based on end conditions are examined: both ends pinned, both ends fixed, one end fixed and one end pinned, one end fixed and one end free. Effective lengths are defined for each case and the corresponding critical buckling loads given. Limitations of Euler's theory are noted. Rankine's empirical formula for calculating ultimate
This document provides information about a Mechanics of Materials course, including:
- The instructor's name and credentials.
- An overview of course contents including stresses, strains, torsion, bending, centroids, and beam deflection methods.
- An introduction to mechanics of materials and the objectives to analyze stresses, strains, and displacements in structures.
- Key terms like stress, strain, axial force, normal force, shear force, deformation, prismatic and non-prismatic bars are defined.
- The stress-strain diagram is discussed and key points like the elastic region, proportional limit, yield point, strain hardening, ultimate stress, and necking are explained.
This document discusses stresses in beams and beam deflection. It covers several methods for analyzing bending stresses and deflection in beams, including: [1] the engineering beam theory relating moment, curvature, and stress; [2] double integration and moment area methods for calculating slope and deflection; and [3] Macaulay's method, which simplifies calculations for beams with eccentric loads. Formulas are provided relating bending moment, shear force, curvature, slope, and deflection. Moment-area theorems are also described for relating bending moment to slope and deflection.
Mohr's circle is a graphical representation of the transformation of stresses on planes at a point in a material. It relates normal and shear stresses on inclined planes to the principal stresses. The circle is centered at the average stress and has a radius equal to the difference between the maximum and minimum principal stresses. Mohr's circle allows determination of stresses on any inclined plane from knowledge of the principal stresses and provides insight into failure conditions of materials.
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.
Objective of the experiment:
1 - Study the relationship between the force (P) and
elongation (ΔL).
2 - Stability and study the relationship between strain (ε)
and stress (σ).
3 - Study the concept of the mechanical properties of solids.
4 - Establish a modulus of elasticity (E)
The document discusses stress and strain in engineering structures. It defines load, stress, strain and different types of each. Stress is the internal resisting force per unit area within a loaded component. Strain is the ratio of dimensional change to original dimension of a loaded body. Loads can be tensile, compressive or shear. Hooke's law states stress is proportional to strain within the elastic limit. The elastic modulus defines this proportionality. A tensile test measures the stress-strain curve, identifying elastic limit and other failure points. Multi-axial stress-strain relationships follow Poisson's ratio definitions.
The document discusses different types of strain energy stored in materials when subjected to loads. It defines strain energy as the work done or energy stored in a body during elastic deformation. The types of strain energy discussed include: elastic strain energy, strain energy due to gradual, sudden, impact, shock and shear loading. Formulas are provided to calculate strain energy due to these different loadings. Examples of calculating strain energy in axially loaded bars and beams subjected to bending and torsional loads are also presented.
Structural Mechanics: Shear stress in Beams (1st-Year)Alessandro Palmeri
- The document discusses shear stress in beams, specifically focusing on Jourawski's formula for calculating shear stress.
- Jourawski's formula provides an approximate solution for the shear stress distribution over a beam cross-section using simple equilibrium considerations.
- The formula can be applied to solid rectangular sections, giving a parabolic shear stress distribution, but provides inaccurate results for T-section and I-section flanges.
- For T-sections and I-sections, the formula can be applied to the web, where it accurately models the shear stress as parabolic. The maximum shear stress occurs at the neutral axis.
This document summarizes concepts related to torsion and the torsion of circular elastic bars. It discusses the assumptions made in analyzing torsion, including that shear strain varies linearly from the central axis. It also covers determining shear stress and torque using the polar moment of inertia for circular cross-sections. The relationships between applied torque, shear stress, shear strain, and angle of twist are defined. Stress concentrations and alternative differential equations approaches are also summarized.
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.
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.
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.
Maximum principal stress theory.
Maximum shear stress theory.
Maximum shear strain theory.
Maximum strain energy theory.
Maximum shear strain energy theory.
This document gives the class notes of Unit 6: Bending and shear Stresses in beams. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
- Stress is defined as force per unit area and can be divided into normal and shear components at a point. Stress around a point in 3D forms a stress ellipsoid with three orthogonal principal stress directions.
- Strain is the change in size and shape of a body due to applied stresses. It includes extension, shear and changes to the ellipsoid shape defined by finite stretches.
- The relationship between stress and strain is evaluated through rock deformation experiments using triaxial apparatus to measure shortening, strain rates, and ductility. The results relate to the rheology and deformation mechanisms in rocks.
This document provides an overview of topics related to strength of materials and mechanics of solids, including normal stress and strain, shear stress and strain, strain energy, impact loads, principal stress and strain, Mohr's stress circle, equilibrium equations, Hooke's law, and theories of failure. It includes definitions, formulas, and examples for each topic.
1. The document discusses unsymmetrical bending of beams. When a beam bends about an axis that is not perpendicular to a plane of symmetry, it is undergoing unsymmetrical bending.
2. Key aspects discussed include determining the principal axes, direct stress distribution, and deflection of beams under unsymmetrical bending. Equations are provided to calculate stresses and deflections.
3. An example problem is given involving finding the stresses at two points on a cantilever beam subjected to an unsymmetrical loading. The principal moments of inertia and neutral axis orientation are calculated.
This document discusses mechanical properties and testing methods. It introduces key terms like stress, strain, tensile testing and how properties like Young's modulus, yield strength and toughness are obtained. Tensile testing provides a stress-strain curve that shows elastic and plastic deformation regions. Ceramics are more brittle so bend testing is used to determine properties like flexural strength. Hardness tests measure a material's resistance to penetration.
Terminology for Mechanical Properties The Tensile Test: Stress-Strain Diagram...manohar3970
Terminology for Mechanical Properties
The Tensile Test: Stress-Strain Diagram
Properties Obtained from a Tensile Test
True Stress and True Strain
The Bend Test for Brittle Materials
Hardness of Materials
This document discusses mechanical properties and tensile testing. It introduces key terms like stress, strain, elastic deformation, plastic deformation, yield strength, tensile strength, and ductility. It explains how mechanical properties like Young's modulus, yield strength, and tensile strength are determined from a stress-strain curve generated through uniaxial tensile testing. It also discusses plastic deformation through dislocation motion, strain hardening, necking, and factors that influence properties like processing methods. True stress and true strain are introduced as alternatives to engineering stress and strain for accounting for changes in cross-sectional area during deformation.
This document summarizes the mechanical properties of materials through stress-strain diagrams. It discusses the differences between stress-strain diagrams for ductile versus brittle materials. For ductile materials, the diagram shows elastic behavior, yielding, strain hardening, necking, and true stress-strain. Brittle materials exhibit no yielding and rupture occurs at a much smaller strain. The document also discusses Hooke's law, Poisson's ratio, axial loading of materials, and provides examples of calculating deformation based on applied loads and material properties.
This document discusses various concepts related to stress and strain. It begins by explaining the three main types of loads - tension, compression, and shear. It then provides diagrams demonstrating these different types of loads. The document goes on to define engineering stress and strain and discuss their units. Several mechanical properties are also defined, including yield strength, ultimate tensile strength, and elongation. Finally, the document discusses various tests used to determine mechanical properties, including tensile, compression, hardness, and impact tests.
This document discusses the mechanical properties of solids. It covers topics such as elasticity, stress and strain, mechanical testing techniques for properties like hardness, ductility, deformation mechanisms, fracture, and fatigue. Elasticity is divided into elastic deformation, where the material returns to its original shape, and plastic deformation, where the shape is permanently changed. Stress is defined as force over cross-sectional area while strain is the change in length over original length. Various mechanical tests are used to characterize properties like hardness, toughness, and ductility. Deformation, fracture, and fatigue failure mechanisms are also examined.
This document discusses mechanical properties that can be determined from tensile and shear tests. It defines key terms like stress, strain, elastic modulus, yield strength, and tensile strength. A typical stress-strain curve is shown and each region is explained. The elastic portion is linear up to the yield point, then the plastic region involves necking and strain hardening until ultimate failure. True stress and strain account for changes in cross-sectional area during deformation. The document also compares properties like ductility and toughness between different materials.
This document discusses various mechanical properties that are important for selecting materials for structural components. It describes different types of mechanical tests like tension, compression, torsion, bending, impact and fatigue tests that are conducted on metal specimens to determine properties like strength, ductility and toughness. Specifically, it outlines the process for a uniaxial tension test including the equipment used, steps to conduct the test, and how to analyze the stress-strain diagram produced. It also discusses factors that influence mechanical properties like temperature, notches, grain size and hardness tests.
Em321 lesson 08b solutions ch6 - mechanical properties of metalsusna12345
This document discusses mechanical properties that can be determined from a stress-strain curve obtained via tensile testing. It defines stress and strain, explains elastic and plastic deformation, and introduces key properties like modulus of elasticity, yield strength, ultimate tensile strength, ductility, toughness, and resilience. An example stress-strain curve is analyzed to find these properties numerically. The document emphasizes that stress-strain curves are commonly used instead of force-displacement plots to characterize materials.
The document discusses stress-strain curves, which plot the stress and strain of a material sample under load. It describes the typical stress-strain behavior of ductile materials like steel and brittle materials like concrete. For ductile materials, the curve shows an elastic region, yield point, strain hardening region, and ultimate strength before failure. The yield point marks the transition between elastic and plastic deformation. The document also discusses factors that influence a material's yield stress, such as temperature and strain rate, and implications for structural engineering like reduced buckling strength after yielding.
This document summarizes key concepts from a chapter on strain in mechanics of materials. It discusses two main types of strain - normal and shear - and how stress and strain define a material's mechanical properties. It then focuses on axial deformation and stress-strain diagrams, describing how tensile tests are conducted to generate these diagrams for materials like steel. Key points on the stress-strain curve are identified, such as proportional limit, yield point, ultimate stress, and how they relate to a material's elastic region and factors of safety in design.
This document provides an overview of strength of materials and introduces key concepts. It discusses stress and strain, ductile and brittle materials, and stress-strain diagrams. Stress is defined as the internal resisting force per unit area acting on a material. Strain is the ratio of change in dimension to the original dimension when a body is subjected to external force. Ductile materials show deformation under stress, while brittle materials do not. The stress-strain diagram shows the relationship between stress and strain for ductile and brittle materials.
The document discusses tensile strength and tensile testing. It defines tensile strength as the maximum stress a material can withstand under tension before necking and breaking. A tensile test measures how a material responds to tensile forces by recording the load and elongation of a test specimen. The results are displayed as a stress-strain curve which can show properties like yield strength, ultimate tensile strength, modulus of elasticity, and ductility measures.
The document discusses tensile strength and tensile testing. It defines tensile strength as the maximum stress a material can withstand under tension before necking and breaking. A tensile test measures how a material responds to tensile forces by recording the load and elongation of a test specimen. The results are used to determine various tensile properties including modulus of elasticity, yield strength, ultimate tensile strength, and measures of ductility. Hooke's law and concepts like strain, stress-strain curves, and necking are also explained in the context of understanding a material's tensile behavior.
The document discusses various mechanical properties of metals including resilience, toughness, true stress and strain, hardening, fatigue failure, creep, and factor of safety.
Resilience is the ability of a material to absorb energy elastically. Toughness is the total energy absorbed before fracture. True stress and strain account for changes in cross-sectional area during deformation. Hardening occurs as yield strength increases with plastic deformation. Fatigue failure results from initiation and propagation of cracks under cyclic stresses. Creep is permanent deformation over time at high temperatures and stresses. Factor of safety determines a system's load-carrying capacity beyond the actual load.
This document provides an introduction to strength of materials, including concepts of stress, strain, Hooke's law, stress-strain relationships, elastic constants, and factors of safety. It defines key terms like stress, strain, elastic limit, modulus of elasticity, and ductile and brittle material behavior. Examples of stress and strain calculations are provided for basic structural elements like rods, bars, and composite structures. The document also covers compound bars, principle of superposition, and effects of temperature changes.
1. The document discusses various mechanical properties including stress, strain, elastic behavior, plastic behavior, toughness, and properties of ceramics, metals, and polymers.
2. Key mechanical properties addressed for materials include yield strength, tensile strength, elastic modulus, ductility, and hardness.
3. The mechanical behavior of different classes of materials like ceramics, metals, and polymers is compared in terms of stress-strain curves and how properties vary with temperature and loading rate.
Structural Integrity Analysis: Chapter 3 Mechanical Properties of MaterialsIgor Kokcharov
Structural Integrity Analysis features a collection of selected topics on structural design, safety, reliability, redundancy, strength, material science, mechanical properties of materials, composite materials, welds, finite element analysis, stress concentration, failure mechanisms and criteria. The engineering approaches focus on understanding and concept visualization rather than theoretical reasoning. The structural engineering profession plays a key role in the assurance of safety of technical systems such as metallic structures, buildings, machines, and transport. The third chapter explains the engineering tests and fundamentals of mechanical properties of materials.
Mechanical properties of materials (lecture+2).pdfHeshamBakr3
The document discusses the mechanical properties of materials when subjected to different types of loading like axial, lateral, and torsional loads. It defines concepts like stress, strain, elastic and plastic deformation. It explains stress-strain diagrams and how they are used to determine properties like modulus of elasticity, yield strength, tensile strength, ductility, toughness, and resilience. Typical stress-strain behaviors of ductile and brittle materials are compared. Examples of determining properties from stress-strain curves are also provided.
Chapter-1 Concept of Stress and Strain.pdfBereketAdugna
The document discusses concepts of stress and strain in materials. It defines stress as an internal force per unit area within a material. Stress can be normal (perpendicular to the surface) or shear (parallel to the surface). Normal stress can be tensile or compressive. Strain is a measure of deformation in response to stress. Hooke's law states that stress is proportional to strain in the elastic region. Poisson's ratio describes the contraction that occurs perpendicular to an applied tensile load. Stress-strain diagrams are used to analyze a material's behavior under different loads. The document also discusses volumetric strain, shear stress and strain, bearing stress, and provides examples of stress and strain calculations.
Similar to Chapter 2: Axial Strains and Deformation in Bars (20)
Chapter 11: Stability of Equilibrium: ColumnsMonark Sutariya
1) The document discusses various buckling modes of columns including flexural, torsional-flexural, and torsional buckling. It provides examples of buckling in thin-walled tubes and prismatic members.
2) Euler buckling formulas are presented for columns with different end conditions, such as both ends pinned, one end fixed and one end pinned. The critical buckling load depends on the effective length which accounts for the end conditions.
3) Limitations of the Euler formulas and generalized formulas are discussed. The tangent modulus formula extends the elastic analysis to the inelastic range by using the tangent modulus.
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.
Chapter 7: Shear Stresses in Beams and Related ProblemsMonark Sutariya
This document discusses shear stresses in beams. It defines shear stress and shear flow, and describes how to calculate them using the shear stress formula. It discusses limitations of this formula and how shear stresses behave in beam flanges and at boundaries. The concept of the shear center is introduced as the point where an applied force will not cause twisting. Methods for combining direct and torsional shear stresses are also covered.
Chapter 6: Pure Bending and Bending with Axial ForcesMonark Sutariya
This summary provides the key points about pure bending and bending with axial forces from the document:
1. Pure bending occurs when a beam segment is in equilibrium under bending moments alone, with examples being a cantilever loaded at the end or a beam segment between concentrated forces.
2. For beams with symmetric cross-sections, plane sections remain plane after bending according to the fundamental flexure theory. The elastic flexure formula gives the normal stress as proportional to the bending moment and the distance from the neutral axis.
3. The second moment of area, or moment of inertia, represents the beam's resistance to bending and is used to calculate maximum bending stresses. The elastic section modulus is a ratio of the moment
Chapter 5: Axial Force, Shear, and Bending MomentMonark Sutariya
1. A beam can experience three internal forces at a section - axial force, shear, and bending moment. Even for planar beams, all three forces may develop.
2. There are three types of supports - roller/link, pin, and fixed. Roller/link supports resist one force, pin supports resist two forces, and fixed supports resist two forces and a moment.
3. Beams can experience different load types - concentrated, uniform distributed, and varying distributed loads. Methods are presented to calculate the shear, axial, and bending effects of these loads on beams.
Chapter 1: Stress, Axial Loads and Safety ConceptsMonark Sutariya
This document provides an overview of stress, axial loads, and safety concepts in mechanics of solids. It discusses general concepts of stress including stress tensors and transformations. It also examines stresses in axially loaded bars, including normal and shear stresses. Finally, it covers deterministic and probabilistic design bases, discussing factors of safety, experimental data distributions, and theoretical probabilistic approaches to structural design and failure.
Cricket management system ptoject report.pdfKamal Acharya
The aim of this project is to provide the complete information of the National and
International statistics. The information is available country wise and player wise. By
entering the data of eachmatch, we can get all type of reports instantly, which will be
useful to call back history of each player. Also the team performance in each match can
be obtained. We can get a report on number of matches, wins and lost.
Better Builder Magazine brings together premium product manufactures and leading builders to create better differentiated homes and buildings that use less energy, save water and reduce our impact on the environment. The magazine is published four times a year.
This is an overview of my career in Aircraft Design and Structures, which I am still trying to post on LinkedIn. Includes my BAE Systems Structural Test roles/ my BAE Systems key design roles and my current work on academic projects.
Sachpazis_Consolidation Settlement Calculation Program-The Python Code and th...Dr.Costas Sachpazis
Consolidation Settlement Calculation Program-The Python Code
By Professor Dr. Costas Sachpazis, Civil Engineer & Geologist
This program calculates the consolidation settlement for a foundation based on soil layer properties and foundation data. It allows users to input multiple soil layers and foundation characteristics to determine the total settlement.
This is an overview of my current metallic design and engineering knowledge base built up over my professional career and two MSc degrees : - MSc in Advanced Manufacturing Technology University of Portsmouth graduated 1st May 1998, and MSc in Aircraft Engineering Cranfield University graduated 8th June 2007.
2. Part A
Strains and Deformations in Axially Loaded Bars
• Gage length = initial distance between two points
• The elongation (ε) per unit of initial gage length is given as,
• Extensional Strain: -
Where, L0 = initial gage length, L = observed length under a given load,
The gage elongation, ΔL = L – L0
2
3. • In some engineering application, as in metal forming, the strains may
be large.
• True or Natural Strain:
𝜀 = 𝐿0
𝐿 𝑑𝐿
𝐿
= ln
𝐿
𝐿0
= ln (1 + ε)
• Natural strains are useful in theories of viscosity and viscoplasticity for
expressing an instantaneous rate of deformation.
• For measuring small strains, we can use expendable electric strain
gages, made of very fine wire or foil that is glued to the member
being investigated. Changing in length alters the electric resistance of
the gage, which can be measured and calibrated to indicate the strain.
• Extensometer can be used for measuring strain.
3
4. Fig. 2: Wire strain gage (excluding top protective cover)
Fig. 1: Diagram of a tension
specimen in a testing machine
4
5. Stress - Strain Relationships
Fig. 3: Typical stress-strain diagrams for different steels
5
6. Fig. 4: Typical stress-strain diagrams for different materials
6
7. • Steel and Aluminum exhibit ductile behavior, and a fracture occurs
only after a considerable amount of deformation.
• These failures occur primarily due to slip in shear along the planes
forming approximately 45° angles with the axis of the load.
• A typical “cup and cone” fracture may be detected at fracture planes.
• In Fig. 5(b) , the upper curve represents some brittle tool steels or
concrete in tension, the middle one of aluminum alloys or plastics,
and the lower curve represents stress- strain characteristics of rubber.
• The terminal point of curve represents complete failure (rupture) of a
specimen.
• Ductile - Materials capable of withstanding large strains without a
significant increase in stress.
• Brittle - converse of Ductile.
7
9. • Some transverse contraction or expansion of a material takes place. In
materials like mild steel or aluminum, near the breaking point
‘necking’ can be observed.
• True Stress = Dividing the applied force at a given point in the test, by
the corresponding actual area of a specimen at the same instant.
• Hooke’s law = Up to some point as A, the relationship between stress
and strain may be said to be linear for all materials.
𝜎 = 𝜀𝐸
Where, E = elastic modulus or young’s modulus.
• The stress corresponds to point A is known as ‘elastic limit’.
• Elastic modulus is a definitive property of material.
• For all steels E at room temperature is between 29 × 106 to 30 × 106
psi, or 200 to 207 GPa.
9
10. • Anisotropic = materials having different physical properties in
different directions, e.g., single crystals and wood.
• The highest point B represents, the ‘ultimate strength’ of a material.
• Stress associate with the long plateau ab, is called the yield strength
of a material.
• At an essentially constant stress, strains 15 to 20 times those that
take place up to the proportional limit occur during yielding.
Fig. 6: Offset method of determining the
Yield strength of a material
10
11. • If in stressing a material its elastic limit is exceeded, on unloading it
usually responds approximately in a linear elastic manner and a
permanent deformation develops at no external load.
• The area enclosed by the loop corresponds to dissipated energy
released through heat.
Fig. 7: stress-strain diagrams: (a) linear elastic material, (b) nonlinear elastic material, (c) plastic material
11
12. • For ductile materials, stress-strain diagram obtained for short
compressions blocks are close to those found in tension.
• Brittle materials, such as cast iron and concrete are very weak in
tension but not in compression.
• Constitutive relations or laws = additional stress-strain relations to
assumption of linearly elastic behavior of material.
• Ideally plastic behavior = a large amount of unbounded deformation
can take place at a constant stress.
• Assumptions: the mechanical properties of the material are the same
in both tension and compression. During unloading, the material
behaves elastically.
• A stress can range and terminate anywhere between + 𝜎 𝑦𝑝 and
− 𝜎 𝑦𝑝.
12
13. Fig. 8: Idealized stress-strain diagrams: (a) rigid perfectly plastic material, (b) elastic - perfectly plastic material,
(c) Elastic – linearly hardening material
• Strain hardening = Beyond the elastic range, on an increase in strain,
many material resists additional stress.
13
14. • Ramberg and Osgood equation:
𝜀
𝜀0
=
𝜎
𝜎0
+
3
7
𝜎
𝜎0
𝑛
• The constants 𝜀0 and 𝜎0 correspond to the yield point, which for all
cases other than that of ideal plasticity, is found by the offset method.
• The exponent ‘n’ determines the shape of the curve.
Fig. 9: Ramberg - Osgood
stress - strain diagrams
14
15. • The Ramberg-Osgood equation is a continuous mathematical
function. An instantaneous or tangent modulus, Et =
𝑑𝜎
𝑑𝜀
can be
uniquely determined.
• A series of characteristic loops, referred as ‘hysteretic loops’
representing the dissipation of energy can be seen in Fig. 10
• Stress- strain diagram is strongly dependent on ambient temperature.
(Fig. 11)
• Creep = with time dependent behavior and a member subjected to a
constant stress, the elongations or deflections continue to increase
with time. (Fig. 12)
• Relaxation = The pre-stress in bolts of mechanical assemblies
operating at high temperature, as well as pre-stress in steel tendons
in reinforced concrete, tend to decrease gradually with time. (Fig. 13)
15
16. Fig. 10: Menegotto–Pinto computer model simulation Fig. 11: Effect of strain rate and temperature on
of cyclic stress-strain diagrams for steel stress-strain curves for 6061-T6 aluminum alloy
16
17. Fig. 12: creep in bar under constant stress Fig. 13: stress-relaxation curve
17
18. Deformation of Axially Loaded Bars
• The deflection characteristics of bars also provide necessary
information for determining the stiffness of systems in mechanical
vibration analysis.
• The normal strain, 𝜀 𝑥 in the x-direction is
𝑑𝑢
𝑑𝑥
, du is the axial
deformation of the infinitesimal element.
• Change in length between 2 points,
∆ = 0
𝐿
𝜀 𝑥 𝑑𝑥 =
𝑃 𝑥
𝐴 𝑥 𝐸 𝑥
𝑑𝑥 (∵ Hooke’s law)
18
19. Poisson’s ratio
• Lateral (transverse) expansion or contraction to the direction of
applied force.
• For elastic, isotropic and homogeneous materials,
𝜈 =
𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
𝑎𝑥𝑖𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
= −
𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
𝑎𝑥𝑖𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
• Here axial strains are caused by uniaxial stress only.
• For some concretes 𝜈 = 0.1, for rubber 𝜈 = 0.5(maximum)
• The Poisson effect exhibited by materials causes no additional
stresses other than compressive or tensile unless the transverse
deformation is inhibited or prevented.
19
20. Thermal Strain and Deformation
• Thermal strain, 𝜀 𝑇 = 𝛼(𝑇 − 𝑇0)
• Where, 𝛼 = coefficient of thermal expansion
• The extensional deformation, ∆ 𝑇= 𝛼(𝑇 − 𝑇0)L
Saint-Venant’s Principle and Stress Concentrations
• In reality, applied forces often approximate concentrated forces, and
the cross section of members can change abruptly. This causes stress
and strain disturbances in the proximity of such forces and changes in
cross-section.
20
21. • The manner of force application on stresses is important only in the
vicinity of the region where the force is applied.
Fig. 14: Stress distribution near a concentrated force in a rectangular elastic plane
21
22. • Using Finite element method, The initial undeformed mesh into
which the planar block is arbitrarily subdivided, and the greatly
exaggerated deformed mesh caused by the applied force are shown
in Fig. 15.
Fig. 15: (a) undeformed and deformed mesh on an elastic plane, (b) 𝜎 𝑦 contours,
(c) Normal stress distributions at b/4 and b/2 below top.
22
23. • At bolt holes or changes in cross section, the maximum normal
stresses are finite, which depends only on the geometric proportions
of a member.
• Stress concentration factor (K) = The ratio of the maximum to the
average stress.
𝜎 𝑚𝑎𝑥 = 𝐾𝜎 𝑎𝑣 = 𝐾
𝑃
𝐴
Fig. 16: Stress concentration factor
23
24. Fig. 17: Stress concentration factors for flat bars in tension
24
25. Elastic Strain Energy for Uniaxial Strain
• Forces and deformations gives ‘internal work’ done in a body by
externally applied forces. It is stored in an elastic body as the ‘internal
elastic energy of dissipation’, or the ‘elastic strain energy’.
Fig. 18: (a) An element in uniaxial tension,
(b) a Hookean stress-strain diagram
25
26. • Strain energy density = strain energy stored in an elastic body per unit
volume of the material, ‘modulus of resilience’
𝑈0 =
𝑑𝑈
𝑑𝑉
=
𝜎 𝑥 𝜀 𝑥
2
=
𝜎 𝑥
2
2𝐸
𝑈 =
𝜎𝑥
2
2𝐸
𝑑𝑉
• Complementary energy = The corresponding area enclosed by the
inclined line and the vertical axis
26
27. • Toughness = material’s ability to absorb energy up to fracture.
• In the inelastic range, only a small part of the energy absorbed by a
material is recoverable. Most of the energy is dissipated in
permanently deforming the material and is lost in heat.
• The energy that may be recovered when a specimen has been
stressed to some point A as in Fig. 19 is represented by the triangle
ABC. Line AB of this triangle is parallel to line OD.
27
29. Deflections by the Energy Method
• We are determining the deflection caused by the application of a
single axial force.
𝑈 = 𝑊𝑒 =
1
2
𝑃∆ =
𝑃2
𝐿
2𝐴𝐸
Dynamic and Impact Loads
• A freely falling weight, or a moving body, that strikes a structure
delivers ‘dynamic’ or ‘impact’ load or force.
29
30. • Idealizing Assumptions:
1. Materials behave elastically, and no dissipation of energy takes
place at the point of impact or at the supports owing to local
inelastic deformation of materials.
2. The inertia of the system resisting an impact may be neglected.
3. The deflection of the system is directly proportional to the
magnitude of the applied force whether a force is dynamically or
statistically applied.
• Consider a weight ‘W’, falling from the height ‘h’ striking a spring
having constant ‘k’.
30
31. Fig. 20: Behavior of an elastic system under an impact force
• Static deflection due to weight w, ∆ 𝑠𝑡=
𝑊
𝑘
• The maximum dynamic deflection, ∆ 𝑚𝑎𝑥=
𝑃 𝑑𝑦𝑛
𝑘
31
32. • Therefore, 𝑃𝑑𝑦𝑛 =
∆ 𝑚𝑎𝑥
∆ 𝑠𝑡
W
• External work can be equated to internal strain energy,
𝑊 ℎ + ∆ 𝑚𝑎𝑥 = 1
2
𝑃𝑑𝑦𝑛∆ 𝑚𝑎𝑥
• Simplifying, ∆ 𝑚𝑎𝑥 = ∆ 𝑠𝑡 1 + 1 +
2ℎ
∆ 𝑠𝑡
and 𝑃𝑑𝑦𝑛 = 𝑊 1 + 1 +
2ℎ
∆ 𝑠𝑡
• If, h is large compared to ∆ 𝑠𝑡, the impact factor is approximately equal
to
2ℎ
∆ 𝑠𝑡
32
33. Part B – Statically Indeterminate System
• Basic Concepts:
1. ‘Equilibrium conditions’ for the system must be assured both in
global and local sense.
2. ‘Geometric compatibility’ among the deformed parts of a body and
at the boundaries must be satisfied.
3. ‘Constitutive (stress-strain) relations’ for the materials of the system
must be complied with.
33
34. Force Method of Analysis
• On applying force P at B, reactions R1 and R2 develop at the ends and
the system deforms.
Fig. 21: Force (flexibility) method of elastic analysis for a statically indeterminate
Axially loaded bar
34
35. • If the flexibility of the lower elastic bar is f2, the deflection
∆0 = 𝑓2 𝑃
∆1 = 𝑓1 + 𝑓2 𝑅1
• The compatibility of deformations at A is then achieved by,
∆0 + ∆1 = 0
𝑅1 = −
𝑓2
𝑓1 + 𝑓2
𝑃
35
36. Introduction to Displacement Method
• The Stiffness, 𝑘𝑖 = 𝐴𝑖 𝐸𝑖 𝐿𝑖
−𝑘1∆ − 𝑘2∆ + 𝑃 = 0
∆ =
𝑃
𝑘1+𝑘2
• The equilibrium conditions for the
free-bodies at i nodes A and C are
𝑅1 = −
𝑘1
𝑘1+ 𝑘2
𝑃
𝑅2 = −
𝑘2
𝑘1+ 𝑘2
𝑃
Fig. 22: Displacement method of analysis for a
Statically indeterminate axially loaded bar
36
37. Displacement Method with Several DoF
• Consider a bar system consisting of three segments of variable
stiffness defined by their respective spring constants ki’s.
• Each node marked in the figure from 1 to 4, is permitted to displace
vertically in either direction. Therefore, 4 DoF (one DoF per node)
• With no deflection at the ends, we have statically indeterminate
problem.
• With the adopted sign convention, the bar segment extension
between the ith and the (i+1)th node is ∆𝑖 − ∆𝑖+1 .
• The internal tensile force = ∆𝑖 − ∆𝑖+1 𝑘𝑖
37
39. • Equilibrium condition 𝐹𝑥 = 0 for each node.
• Node 1:
• Recasting these equations into the following form,
• We know either the deflections ∆𝑖’s or reactions Pi’s.
39
40. • Writing these equations in the matrix form,
• This is symmetric stiffness matrix. We can use computer programs for
solving these equations simultaneously.
40
41. Statically Indeterminate Non-linear problems
• Global equilibrium equation:
R1 + R2 + P = 0
• Compatibility at the juncture of two bar
segments. Ends A and C are held, Fig. 24: A bar of
nonlinear material
∆ 𝐴𝐵= −∆ 𝐵𝐶
• Using constitutive law, If the bar behavior is
linearly elastic,
𝑅1 𝐿1
𝐴1 𝐸1
= −
𝑅2 𝐿2
𝐴2 𝐸2
41
42. Alternative Differential Equation Approach for
Deflections
• The axial deflection u of a bar is determined by solving a first-order
differential, 𝜀 𝑥 =
𝑑𝑢
𝑑𝑥
=
𝜎
𝐸
=
𝑃
𝐴𝐸
𝑃 = 𝐴𝐸
𝑑𝑢
𝑑𝑥
• Consider a system as shown in Fig. 25
Fig. 25: Infinitesimal element of an axially loaded bar
42
43. 𝐹𝑥 = 0,
𝑑𝑃 + 𝑝 𝑥 𝑑𝑥 = 0
𝑑𝑃
𝑑𝑥
= −𝑝 𝑥
• The rate of change with x of the internal axial force P is equal to
negative of the applied force px.
• Now, assuming AE as constant,
𝑑
𝑑𝑥
𝑑𝑢
𝑑𝑥
=
1
𝐴𝐸
𝑑𝑃
𝑑𝑥
𝐴𝐸
𝑑2 𝑢
𝑑𝑥2
= −𝑝 𝑥
43