The document discusses plate bending theory and the stress-strain hypothesis. Some key points:
1) Plate bending theory studies the behavior of thin, flat plates under external loads and is based on the assumption that plates are very thin compared to their other dimensions and undergo small deformations.
2) The stress-strain hypothesis relates the stress in a plate to the strain it undergoes, assuming stress is proportional to strain via the modulus of elasticity. It also assumes normal stress is proportional to curvature and shear stress is proportional to the rate of change of curvature.
3) The stress-strain hypothesis allows derivation of equations relating external loads on a plate to resulting deformations and stresses, using the principle of virtual work
This document provides an introduction to the theory of plates, which are structural elements that are thin and flat. It defines what is meant by a thin plate and discusses different plate classifications based on thickness. The document derives the basic equations that describe plate behavior by taking advantage of the plate's thin, planar character. It also discusses three-dimensional considerations like stress components, equilibrium, strain and displacement for putting the plate theory into context.
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 the concept of shear center for beams with non-symmetric cross sections. It defines shear center as the point where a load can be applied such that the beam only bends with no twisting. Formulas to calculate the shear center are presented for common cross sections like channels, I-beams, and circular tubes. Examples of determining the shear center for different cross sections are included. The importance of applying loads through the shear center to prevent twisting is emphasized.
This document summarizes chapter 7 from the textbook "Mechanics of Materials" which discusses transformations of stress and strain. It introduces the general state of stress as defined by 6 stress components and explains plane stress as a simplified state. Plane stress is further analyzed using Mohr's circle to determine principal stresses and maximum shear stress. Several examples are provided to demonstrate applying Mohr's circle to calculate stresses under different loading conditions.
Columns are structural members that experience compression loads. They can buckle if loaded beyond their buckling (or critical) load. Short columns fail through crushing, while long columns fail through lateral buckling. The Euler formula calculates the buckling load of a long column based on its properties and end conditions. The Rankine-Gordon formula provides a more accurate calculation of buckling load that applies to all column types by accounting for both buckling and crushing. Proper design of columns involves ensuring they are loaded below their safe loads, which incorporate factors of safety applied to the theoretical buckling loads.
1. Strain energy is the energy stored in a body when it is strained within the elastic limit due to an applied load. The formula for strain energy is U = σ2/2E x V, where U is strain energy, σ is stress, E is modulus of elasticity, and V is volume.
2. Resilience is the total strain energy stored in a body within the elastic limit. Proof resilience is the maximum strain energy that can be stored at the elastic limit. Modulus of resilience is the maximum strain energy that can be stored per unit volume at the elastic limit.
3. An impact or shock load is a sudden load applied to a body, such as a load falling from
This document provides an overview of plate bending theory. It discusses how plate theory models bending in thin plates using Kirchhoff's plate theory and thick plates using Mindlin plate theory. Kirchhoff's plate theory assumes plates bend without shear deformation and strains are related to plate deflection. Mindlin plate theory allows for shear deformation and separates plate rotation and deflection. The document also discusses deriving plate stresses and forces, applying boundary conditions, and using triangular plate bending elements in finite element analysis.
The document discusses stress and strain under axial loading. It covers topics such as normal strain, stress-strain diagrams, Hooke's law, elastic and plastic behavior, fatigue, deformations under axial loading, static indeterminacy, thermal stresses, Poisson's ratio, generalized Hooke's law, shear strain, relations among elastic properties, composite materials, stress concentrations, and examples.
This document provides an introduction to the theory of plates, which are structural elements that are thin and flat. It defines what is meant by a thin plate and discusses different plate classifications based on thickness. The document derives the basic equations that describe plate behavior by taking advantage of the plate's thin, planar character. It also discusses three-dimensional considerations like stress components, equilibrium, strain and displacement for putting the plate theory into context.
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 the concept of shear center for beams with non-symmetric cross sections. It defines shear center as the point where a load can be applied such that the beam only bends with no twisting. Formulas to calculate the shear center are presented for common cross sections like channels, I-beams, and circular tubes. Examples of determining the shear center for different cross sections are included. The importance of applying loads through the shear center to prevent twisting is emphasized.
This document summarizes chapter 7 from the textbook "Mechanics of Materials" which discusses transformations of stress and strain. It introduces the general state of stress as defined by 6 stress components and explains plane stress as a simplified state. Plane stress is further analyzed using Mohr's circle to determine principal stresses and maximum shear stress. Several examples are provided to demonstrate applying Mohr's circle to calculate stresses under different loading conditions.
Columns are structural members that experience compression loads. They can buckle if loaded beyond their buckling (or critical) load. Short columns fail through crushing, while long columns fail through lateral buckling. The Euler formula calculates the buckling load of a long column based on its properties and end conditions. The Rankine-Gordon formula provides a more accurate calculation of buckling load that applies to all column types by accounting for both buckling and crushing. Proper design of columns involves ensuring they are loaded below their safe loads, which incorporate factors of safety applied to the theoretical buckling loads.
1. Strain energy is the energy stored in a body when it is strained within the elastic limit due to an applied load. The formula for strain energy is U = σ2/2E x V, where U is strain energy, σ is stress, E is modulus of elasticity, and V is volume.
2. Resilience is the total strain energy stored in a body within the elastic limit. Proof resilience is the maximum strain energy that can be stored at the elastic limit. Modulus of resilience is the maximum strain energy that can be stored per unit volume at the elastic limit.
3. An impact or shock load is a sudden load applied to a body, such as a load falling from
This document provides an overview of plate bending theory. It discusses how plate theory models bending in thin plates using Kirchhoff's plate theory and thick plates using Mindlin plate theory. Kirchhoff's plate theory assumes plates bend without shear deformation and strains are related to plate deflection. Mindlin plate theory allows for shear deformation and separates plate rotation and deflection. The document also discusses deriving plate stresses and forces, applying boundary conditions, and using triangular plate bending elements in finite element analysis.
The document discusses stress and strain under axial loading. It covers topics such as normal strain, stress-strain diagrams, Hooke's law, elastic and plastic behavior, fatigue, deformations under axial loading, static indeterminacy, thermal stresses, Poisson's ratio, generalized Hooke's law, shear strain, relations among elastic properties, composite materials, stress concentrations, and examples.
This document provides information about shear stresses and shear force in structures. It includes:
- Definitions of shear force and shear stress. Shear force is an unbalanced force parallel to a cross-section, and shear stress develops to resist the shear force.
- Explanations of horizontal and vertical shear stresses that develop in beams due to bending moments. Shear stress is highest at the neutral axis and reduces towards the top and bottom of the beam cross-section.
- Derivations of formulas for calculating shear stress across different beam cross-sections. Shear stress is directly proportional to the shear force and beam geometry.
- Examples of calculating maximum and average shear stresses for various cross-sections
This document provides an overview of the theory of elasticity. It discusses three key topics:
1) Stress and strain analysis including three-dimensional stress and strain, stress-strain transformations, stress invariants, and equilibrium and compatibility equations.
2) Two-dimensional problems involving solutions in Cartesian and polar coordinates, as well as beam bending problems.
3) Energy principles, variational methods, and numerical methods for solving elasticity problems.
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.
- This document discusses a teaching schedule for a course on the failure of slender and stocky columns. It covers topics like column stability, unsymmetric bending, and complex stress/strain over 11 weeks.
- The key learning outcomes are to derive the Euler critical load for slender pinned-pinned columns under compression and to predict the failure mode of short and slender columns.
- The document motivates the importance of considering both the stiffness and strength of materials, and how the slenderness of a column affects its failure mode in compression.
This document provides an overview of basic equations for the theory of plates and shells. It discusses the state of stress and strain at a point, including defining the six independent stress and strain components. It presents the relationships between strain and displacement, and discusses the equilibrium equations relating stress and body forces. Finally, it provides the equations for both Cartesian and cylindrical coordinate systems. The key concepts covered are the fundamental equations that form the basis of plate and shell theory.
The document discusses plane stress and plane strain models. Plane stress deals with thin slabs where the thickness is much smaller than the in-plane dimensions, resulting in zero stresses in the thickness direction and no variation through the thickness. Plane strain deals with long prismatic bodies, where the length is much greater than the in-plane dimensions, resulting in zero strains in the length direction. Both make assumptions about stress and strain variations to reduce the equations to a 2D form, but these are approximations as there are actually non-zero secondary stresses and strains ignored in the models.
This document summarizes key concepts from Chapter 7 of the textbook "Mechanics of Materials" related to transformations of stress and strain. It introduces plane stress and strain, principal stresses and strains, Mohr's circle representation for analyzing stresses and strains under rotations. It provides examples and sample problems demonstrating how to determine principal stresses and strains, maximum shearing stresses, and stress/strain components under rotated reference frames using Mohr's circle. Diagrams and equations are presented for common stress/strain transformations.
This document summarizes the seminar work on the analysis and design of reinforced concrete curved beams. It discusses that curved beams experience both bending moments and torsional moments due to loads acting outside the line of supports. The document outlines the methodology used, which includes manual design using limit state method according to Indian code IS 456 and software analysis and design using ETABS. It presents the important equations for calculating bending moments and torsion in circular beams. A design example is included to demonstrate and compare the manual and software based designs. The conclusion indicates that manual design considers the combined effect of bending and twisting better than software.
Finite Element analysis -Plate ,shell skew plate S.DHARANI KUMAR
This document provides an overview of plate and shell theory and finite element analysis for plates and shells. It discusses the assumptions and applications of thin plate theory, thick plate theory, and shell theory. It also describes different types of finite elements that can be used to model plates and shells, including plate, shell, solid shell, curved shell, and degenerated shell elements. Additionally, it covers skew plates and different discretization methods that can be used for finite element analysis of skew plates.
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
This document discusses curved beams and provides equations for calculating stresses in curved beams. It begins by stating that beam theory can be applied to curved beams to determine stresses in shapes like crane hooks. It provides symbols for variables used in the equations. The main differences between straight and curved beams are that the neutral axis and centroid axis do not coincide for curved beams. Equations are provided to calculate strain and stress at different radii along the curved beam based on the eccentricity between the neutral and centroid axes. An example calculation for a crane hook is also shown.
Columns are structural elements that transmit loads in compression from beams and slabs above to other elements below. Columns can experience both axial compression and bending loads. Biaxial bending occurs when a column experiences simultaneous bending about both principal axes, such as in corner columns of buildings. The biaxial bending method permits analysis of rectangular columns under these conditions. The document provides details on analyzing a sample reinforced concrete column for adequacy using the reciprocal load method to check that factored loads do not exceed design capacity. Diagrams are presented showing interaction surfaces and stress distributions for concentrically and eccentrically loaded columns.
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.
Pias Chakraborty presented on the topic of shear stress for their 4th year, 2nd semester Pre-stressed Concrete Lab course taught by Sabreena Nasrin Madam and Munshi Galib Muktadir Sir. Shear stress acts parallel to the selected plane and is determined by the formula tau = F/A, where tau is the shear stress, F is the applied force, and A is the cross-sectional area. Shear stress causes a material to deform into a parallelogram shape and is maximum at the neutral axis of a beam.
Cable Layout, Continuous Beam & Load Balancing MethodMd Tanvir Alam
This document provides information on cable layout and load balancing methods for prestressed concrete beams. It discusses layouts for simple, continuous, and cantilever beams. For simple beams, it describes layouts for pretensioned and post-tensioned beams, including straight, curved, and bent cable configurations. It also compares the load carrying capacities of simple and continuous beams. The document concludes by explaining the load balancing method for design, using examples of how to balance loads in simple, cantilever, and continuous beam configurations.
This document provides an overview of laminated composite materials and refined plate theories used to model their behavior. It discusses how classical plate theory (CPT) and first-order shear deformation theory (FSDT) have limitations for thick laminated composites due to neglecting transverse shear effects. Higher-order theories like trigonometric shear deformation theory (TSDT), hyperbolic shear deformation theory (HSDT), and second-order shear deformation theory (SSDT) are introduced to address these limitations. The objectives of the study are to develop new refined theories, establish their credibility by applying them to static flexure problems, and obtain results for laminated beams and plates under various loadings not widely available in literature.
This document provides information about shear stresses and shear force in structures. It includes:
- Definitions of shear force and shear stress. Shear force is an unbalanced force parallel to a cross-section, and shear stress develops to resist the shear force.
- Explanations of horizontal and vertical shear stresses that develop in beams due to bending moments. Shear stress is highest at the neutral axis and reduces towards the top and bottom of the beam cross-section.
- Derivations of formulas for calculating shear stress across different beam cross-sections. Shear stress is directly proportional to the shear force and beam geometry.
- Examples of calculating maximum and average shear stresses for various cross-sections
This document provides an overview of the theory of elasticity. It discusses three key topics:
1) Stress and strain analysis including three-dimensional stress and strain, stress-strain transformations, stress invariants, and equilibrium and compatibility equations.
2) Two-dimensional problems involving solutions in Cartesian and polar coordinates, as well as beam bending problems.
3) Energy principles, variational methods, and numerical methods for solving elasticity problems.
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.
- This document discusses a teaching schedule for a course on the failure of slender and stocky columns. It covers topics like column stability, unsymmetric bending, and complex stress/strain over 11 weeks.
- The key learning outcomes are to derive the Euler critical load for slender pinned-pinned columns under compression and to predict the failure mode of short and slender columns.
- The document motivates the importance of considering both the stiffness and strength of materials, and how the slenderness of a column affects its failure mode in compression.
This document provides an overview of basic equations for the theory of plates and shells. It discusses the state of stress and strain at a point, including defining the six independent stress and strain components. It presents the relationships between strain and displacement, and discusses the equilibrium equations relating stress and body forces. Finally, it provides the equations for both Cartesian and cylindrical coordinate systems. The key concepts covered are the fundamental equations that form the basis of plate and shell theory.
The document discusses plane stress and plane strain models. Plane stress deals with thin slabs where the thickness is much smaller than the in-plane dimensions, resulting in zero stresses in the thickness direction and no variation through the thickness. Plane strain deals with long prismatic bodies, where the length is much greater than the in-plane dimensions, resulting in zero strains in the length direction. Both make assumptions about stress and strain variations to reduce the equations to a 2D form, but these are approximations as there are actually non-zero secondary stresses and strains ignored in the models.
This document summarizes key concepts from Chapter 7 of the textbook "Mechanics of Materials" related to transformations of stress and strain. It introduces plane stress and strain, principal stresses and strains, Mohr's circle representation for analyzing stresses and strains under rotations. It provides examples and sample problems demonstrating how to determine principal stresses and strains, maximum shearing stresses, and stress/strain components under rotated reference frames using Mohr's circle. Diagrams and equations are presented for common stress/strain transformations.
This document summarizes the seminar work on the analysis and design of reinforced concrete curved beams. It discusses that curved beams experience both bending moments and torsional moments due to loads acting outside the line of supports. The document outlines the methodology used, which includes manual design using limit state method according to Indian code IS 456 and software analysis and design using ETABS. It presents the important equations for calculating bending moments and torsion in circular beams. A design example is included to demonstrate and compare the manual and software based designs. The conclusion indicates that manual design considers the combined effect of bending and twisting better than software.
Finite Element analysis -Plate ,shell skew plate S.DHARANI KUMAR
This document provides an overview of plate and shell theory and finite element analysis for plates and shells. It discusses the assumptions and applications of thin plate theory, thick plate theory, and shell theory. It also describes different types of finite elements that can be used to model plates and shells, including plate, shell, solid shell, curved shell, and degenerated shell elements. Additionally, it covers skew plates and different discretization methods that can be used for finite element analysis of skew plates.
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
This document discusses curved beams and provides equations for calculating stresses in curved beams. It begins by stating that beam theory can be applied to curved beams to determine stresses in shapes like crane hooks. It provides symbols for variables used in the equations. The main differences between straight and curved beams are that the neutral axis and centroid axis do not coincide for curved beams. Equations are provided to calculate strain and stress at different radii along the curved beam based on the eccentricity between the neutral and centroid axes. An example calculation for a crane hook is also shown.
Columns are structural elements that transmit loads in compression from beams and slabs above to other elements below. Columns can experience both axial compression and bending loads. Biaxial bending occurs when a column experiences simultaneous bending about both principal axes, such as in corner columns of buildings. The biaxial bending method permits analysis of rectangular columns under these conditions. The document provides details on analyzing a sample reinforced concrete column for adequacy using the reciprocal load method to check that factored loads do not exceed design capacity. Diagrams are presented showing interaction surfaces and stress distributions for concentrically and eccentrically loaded columns.
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.
Pias Chakraborty presented on the topic of shear stress for their 4th year, 2nd semester Pre-stressed Concrete Lab course taught by Sabreena Nasrin Madam and Munshi Galib Muktadir Sir. Shear stress acts parallel to the selected plane and is determined by the formula tau = F/A, where tau is the shear stress, F is the applied force, and A is the cross-sectional area. Shear stress causes a material to deform into a parallelogram shape and is maximum at the neutral axis of a beam.
Cable Layout, Continuous Beam & Load Balancing MethodMd Tanvir Alam
This document provides information on cable layout and load balancing methods for prestressed concrete beams. It discusses layouts for simple, continuous, and cantilever beams. For simple beams, it describes layouts for pretensioned and post-tensioned beams, including straight, curved, and bent cable configurations. It also compares the load carrying capacities of simple and continuous beams. The document concludes by explaining the load balancing method for design, using examples of how to balance loads in simple, cantilever, and continuous beam configurations.
This document provides an overview of laminated composite materials and refined plate theories used to model their behavior. It discusses how classical plate theory (CPT) and first-order shear deformation theory (FSDT) have limitations for thick laminated composites due to neglecting transverse shear effects. Higher-order theories like trigonometric shear deformation theory (TSDT), hyperbolic shear deformation theory (HSDT), and second-order shear deformation theory (SSDT) are introduced to address these limitations. The objectives of the study are to develop new refined theories, establish their credibility by applying them to static flexure problems, and obtain results for laminated beams and plates under various loadings not widely available in literature.
This document contains information about a teaching schedule for a course on complex stresses. It will cover topics like beam shear stresses, shear centres, virtual forces, compatibility methods, moment distribution methods, column stability, unsymmetric bending, and complex stress/strain over 11 weeks. The lectures and tutorials will be led by various staff members. The document also provides motivations for studying complex stresses, which include the fact that failure often results from different stresses acting together, and discussing examples like welded connections, reinforced concrete, and concrete cylinder tests.
Principle of Virtual Work in structural analysisMahdi Damghani
The document provides an overview of the principle of virtual work (PVW) for structural analysis. Some key points:
1) PVW is based on the concept of work and energy methods. It states that for a structure in equilibrium under applied forces, the total virtual work done by these forces due to a small arbitrary displacement is zero.
2) PVW can be used to determine unknown internal forces or displacements in statically indeterminate structures by applying virtual displacements or forces.
3) Examples demonstrate using PVW to calculate the bending moment at a point in a beam and the force in a member of an indeterminate truss by equating the external virtual work to internal virtual work.
Mohamad Redhwan Abd Aziz is a lecturer at the DEAN CENTER OF HND STUDIES who teaches the subject of Solid Mechanics (BME 2023). The 3 credit hour course involves 2 hours of lectures and 2 hours of labs/tutorials each week. Student assessment includes quizzes, assignments, tests, lab reports, and a final exam. The course objectives are to understand stress, strain, and forces in solid bodies through various principles and experiments. Topic areas covered include stress and strain, elasticity, shear, torsion, bending, deflection, and more. References for the course are provided.
This document discusses analyzing structures with floating or ambiguous supports using virtual supports. It begins by introducing the concept of virtual supports, which allow analyzing structures without fixed coordinates by introducing additional hypothetical supports that do not influence stresses. The document then provides equations for static stress-strain analysis and discusses using virtual supports to determine displacements without altering stresses. As an example, it analyzes a semi-floating roof structure using virtual supports in ABAQUS software. Finally, it notes dynamic systems can be modeled statically using virtual supports by treating inertial forces as external loads.
Types of stresses and theories of failure (machine design & industrial drafti...Digvijaysinh Gohil
This document summarizes different types of stresses and theories of failure in mechanical components. It discusses eight types of stresses: tensile, compressive, bending, direct shear, torsional shear, bearing pressure, crushing, and contact stresses. It then explains three main theories of failure - maximum principal stress theory, maximum shear stress theory, and distortion energy theory - and their applications based on the material properties.
- The document discusses the principle of virtual work, which states that if a system of bodies is in static equilibrium, then the total virtual work done by all active forces for any virtual displacement from the equilibrium position is zero.
- It defines virtual work and explains how to calculate virtual work done by internal forces like axial forces, shear forces, bending moments, and torsion. This includes using linear elastic relationships.
- It provides examples of using the principle of virtual work to determine unknown forces and displacements in statically indeterminate structures. The virtual displacements allow writing equations relating internal and external work to solve for unknowns.
This document provides an overview of beam and column design concepts. It discusses types of beam supports, beams, shear force and bending moment diagrams, stresses in beams from bending and shear, and beam deflection calculations. It also covers column buckling, including the Euler buckling formula and Johnson's equation. The document provides examples of calculating stresses, strains, deflections, and buckling loads for different beam and column scenarios.
ME 205- Chapter 6 - Pure Bending of Beams.pdfaae4149584
This chapter discusses pure bending of beams. Beams are members that support loads applied perpendicular to their longitudinal axis. The objectives are to determine stresses caused by bending, understand bending theory and its applications, and determine normal stresses in symmetric bending of beams. Bending stress depends on the beam cross-section and its properties. Bending causes compression on one face and tension on the other, and causes the beam to deflect. Various beam cross-sections are discussed including symmetric and asymmetric cross-sections. The theory of simple bending is presented including assumptions, strain and stress distributions, and calculations for maximum bending stress using section modulus. Examples are provided to calculate stresses and deformations in beams subjected to bending moments.
This document summarizes key concepts in strength of materials including:
- Analysis of pure bending and symmetrical sections bending in a plane of symmetry
- Skew loading and bending about axes other than axes of symmetry
- Eccentric loading introducing both direct stress and bending stress
- The middle third rule and middle quarter rule defining safe load application areas to avoid tension
Principle of virtual work and unit load methodMahdi Damghani
The document summarizes the principle of virtual work (PVW) which is a fundamental tool in analytical mechanics. It defines virtual work as the work done by a real force moving through an arbitrary virtual displacement. The PVW states that if a particle is in equilibrium, the total virtual work done by the applied forces equals zero. Examples are provided to demonstrate how PVW can be used to determine unknown internal forces and slopes by equating the virtual work of external and internal forces.
The document summarizes the principle of virtual work (PVW) used in structural analysis. It defines work done by forces and virtual work done by internal forces in structures during small imaginary displacements. It provides examples of using PVW to determine bending moment, axial force, and slope of indeterminate structures by equating virtual work done by internal and external forces. The key concepts are defined in less than 3 sentences.
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.
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.
Kantorovich-Vlasov Method for Simply Supported Rectangular Plates under Unifo...IJCMESJOURNAL
In this study, the Kantorovich-Vlasov method has been applied to the flexural analysis of simply supported Kirchhoff plates under transverse uniformly distributed load on the entire plate domain. Vlasov method was used to construct the coordinate functions in the x direction and the Kantorovich method was used to consider the assumed displacement field over the plate. The total potential energy functional and the corresponding Euler-Lagrange equations were obtained. This was solved subject to the boundary conditions to obtain the displacement field over the plate. Bending moments were then obtained using the moment curvature equations. The solutions obtained were rapidly convergent series for deflection, and bending moments. Maximum deflection and maximum bending moments occurred at the center and were also obtained as rapidly convergent series. The series were computed for varying plate aspect ratios. The results were identical with Levy-Nadai solutions for the same problem.
This document provides an introduction and overview of mechanics of materials. It defines key terms like stress, strain, normal stress, shear stress, factor of safety, and allowable stress. It also gives examples of calculating stresses in structural members subjected to various loads. The document is an introductory reading for a mechanics of materials course that will analyze the relationship between external forces and internal stresses and strains in structural elements.
Online train ticket booking system project.pdfKamal Acharya
Rail transport is one of the important modes of transport in India. Now a days we
see that there are railways that are present for the long as well as short distance
travelling which makes the life of the people easier. When compared to other
means of transport, a railway is the cheapest means of transport. The maintenance
of the railway database also plays a major role in the smooth running of this
system. The Online Train Ticket Management System will help in reserving the
tickets of the railways to travel from a particular source to the destination.
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.
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.
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2. Plate Bending Theory
• Plate bending theory is a branch of mechanics that
studies the behavior of thin, flat plates under external
loads.
• The theory is based on the assumption that the plate is
very thin compared to its other dimensions and that it
undergoes small deformations.
3. Stress-Strain Hypothesis
• The stress-strain hypothesis is one of the key concepts
in plate bending theory.
• It relates the stress in the plate to the strain it
undergoes.
• This hypothesis assumes that the stress in the plate is
proportional to its strain, and that the proportionality
factor, known as the modulus of elasticity, is constant.
4. Stress-Strain Hypothesis
• More specifically, the stress-strain hypothesis in plate
bending theory assumes that the normal stress in the
plate is proportional to its curvature.
• And the shear stress is proportional to the rate of
change of curvature.
5. Stress-Strain Hypothesis
• This can be expressed mathematically using the
following equations:
σx = E(y'' + νxy')
τxy = E(1 - ν)x'y'
• where σx is the normal stress in the x-direction, τxy is
the shear stress in the xy-plane, E is the modulus of
elasticity of the plate, ν is Poisson's ratio, and y is the
displacement of the plate in the y-direction.
6. Stress-Strain Hypothesis
• The equations above describe the stress-strain
relationship in the plate, which allows us to derive
equations that relate the external loads applied to the
plate to the resulting deformations and stresses.
• These equations can be derived using the principle of
virtual work, which states that the work done by
external forces on a system is equal to the work done
by the internal forces within the system.
7. Stress-Strain Hypothesis
• It is used in the analysis and design of a wide range of
engineering structures, such as bridges, ships, and
aircraft.
• By understanding the behavior of thin plates under
external loads, engineers can design more efficient and
reliable structures that can withstand the stresses and
strains that they will be subjected to in service.
8. PURBANCHAL UNIVERSITY
KHWOPA ENGINEERING COLLEGE
SOLID MECHANICS
PRINCIPLE OF VIRTUAL WORK
REENA SUWAL (ME07814)
SACHIN POKHAREL (ME07815)
SAMI DANGOL (ME07816)
SUNDAR BARTAULA (ME07817)
Assoc. Prof. Dr. Manjip Shakya
Department of Earthquake Engineering
8th March 2023
9. Introduction
Virtual displacement and virtual Work
• Virtual work is the work done by a real force acting through a virtual displacement or a
virtual force acting through a real displacement.
• A virtual displacement is any displacement consistent with the constraints of the
structure, i.e., that satisfy the boundary conditions at the supports.
• A virtual force is any system of forces in equilibrium.
Principle of virtual Work
The principle of virtual work states that in equilibrium the virtual work of the forces
applied to a system is zero. Newton's laws state that at equilibrium the applied forces are
equal and opposite to the reaction, or constraint forces. This means the virtual work of the
constraint forces must be zero as well.
10. F1 = 1
F1
F3
F2 M1
M2
M3
F2 = 2
F3 = 3
Virtual linear displacement
Virtual Work
15. Mathematical expression
• Consider an elastic system subjected to a number of forces (including
moments) F1, F2, . . . , etc. Let 𝛿1, 𝛿2, . . ., etc. be the corresponding
displacements.
• These are the work absorbing components (linear and angular
displacements) in the corresponding directions of the force as shown
in figure.
16. • Let one of the displacements 𝛿1 be increased by a small quantity
∆𝛿1. During this additional displacement, all other displacements
where forces are acting are held fixed, which means that additional
forces may be necessary to maintain such a condition.
• Further, the small displacement ∆𝛿 1 that is imposed must be
consistent with the constraints acting. For example, if point ‘I’is
constrained in such a manner that it can move only in a particular
direction, then ∆𝛿1 must be consistent with such a constraint.
• A hypothetical displacement of such a kind is called a virtual
displacement. In applying this virtual displacement, the forces F1, F2,
. . ., etc. (except F1) do no work at all because their points of
application do not move (at least in the work-absorbing direction).
• The only force doing work is F1 by an amount F1 ∆𝛿1. plus a fraction
of∆F1 ∆𝛿1. , caused by the change in F1. This additional work is
stored in strain energy ∆𝑈.
18. The work done by the forces must be equal to the strain energy
which is stored up in the system. This fact can be expressed in
another form known as the principle of virtual work, i.e.
19. • In the equations above,
V denotes the material domain,
S the surface completely enclosing V,
and the variational symbol signifies a virtual quantity
upon applying the divergence theorem
➔
20. We know, WE = WI
Substituting the value of WE and WI
………(i)
………(ii) a
………(ii) b
Eqn (i) is a field eqn
Eqn (ii) specify boundary conditions.
21. Conclusion
This principle is applicable to any elastic body
• linear elastic materials
• Non-linear elastic materials
24. Boundary Condition
In physics and engineering, boundary conditions are the set of
conditions that must be satisfied at the edges or boundaries of a
physical system.
Boundary conditions can take many forms, depending on the type
of system being analyzed.
For example, in a heat transfer problem, boundary conditions
might specify the temperature or heat flux at the boundaries of a
system.
In a fluid dynamics problem, boundary conditions might specify
the velocity or pressure at the boundaries of a system.
25. Boundary Condition for Plate Theory
❑ These boundary conditions are important because they
define the support and loading conditions that the
plate is subjected to, and therefore they determine
the deformation and stress distribution in the plate.
❑ In plate bending theory, the boundary conditions
depend on the type of support at the edges of the
plate.
26. Theory of Thin Plate Bending
Equilibrium Equation along z-
Direction:
32. Free Edges:
Moment Mxy and Shear Forces
Q + Q ’
∎
The net force acting on the
face
Q’x=-Mxy=
𝜕Mxy
𝜕𝑦
+Mxy
Q’x=-
𝜕Mxy
𝜕𝑦
Total shear force
Vx=Qx+Q’x
36. Plates :-
▪ body whose lateral dimensions are large compared to the separation
between the surfaces.
▪ are initially flat structural elements.
▪ are subjected to traverse loads (that are
normal to its mid-surface) supported by
bending and shear action.
▪ Thin plates – t < 20b where, b = smallest side
Thick plates – t > 20b
▪ Small deflection – w ≤ t/5
▪ Thin plate theory – Kirchhoff's Classical Plate Theory (KCPT)
Thick plate theory – Reissner - Mindlin Plate Theory (MPT)
2
37. Assumptions :-
▪ Thickness is smaller than other dimensions
▪ Normal stresses in transverse direction are small compared with other
stresses, so neglected
▪ Governing equation is based on undeformed geometry
▪ Material is linearly-elastic body, thus follows Hooke’s law
▪ Middle surface remains unstrained during bending , so taken as neutral
surface
▪ Normal to mid-surface before deformation remain normal to the same
surface after deformation
▪ Traverse shear strains are negligible, doesn’t apply shear across section
is zero.Traverse shear strain have negligible contribution to
deformations.
3
40. 𝜵𝟒
𝒘 =
𝒒
𝑫
→ 𝐃 𝜵𝟒
𝒘(𝒙, 𝒚) = 𝒒(𝒙, 𝒚)
▪ 𝜵𝟒𝒘 is known as Laplace Operator or Biharmonic Operator.This expression is
the basic differential equation for plate bending theory.
▪ This plate equation solution requires specification of appropriate boundary
condition that involves deflection & its derivatives at the edges of the plate.
▪ Kirchhoff's plate equation is widely used model in plate bending theory & has
numerous applications in engineering & physics such as, design of aircraft wings
and analysis of seismic waves in the earth’s crust.
6
43. Plate
A plate is a flat structural element for which the thickness is small compared with the surface
dimensions.
The thickness is usually constant but may be variable and is measured normal to the middle surface
of the plate.
44. Basic theory of thin plates
Assumptions:
One dimension (thickness) is much smaller than the other two dimensions
(width and length) of the plate i.e.; t << Lx, Ly
Shear stress is small; shear strains are small i.e; σz = 0; εz = εxz = εyz = 0
Thin plates must be thin enough to have small shear deformations but thick
enough to accommodate in-plane/membrane forces.
45. Assumptions of Plate Theory
Let the plate mid-surface lie in the x y plane and the z – axis be along the thickness
direction, forming a right handed set, Fig. 6.1.4.
The stress components acting on a typical element of the plate are shown in Fig. 6.1.5.
46. Assumptions of Plate Theory
The following assumptions are made:
(i) The mid-plane is a “neutral plane”
❖ The middle plane of the plate remains free of in-plane stress/strain. Bending of
the plate will cause material above and below this mid-plane to deform in-plane.
❖ The mid-plane plays the same role in plate theory as the neutral axis does in the
beam theory.
(ii) Line elements remain normal to the mid-plane
❖ Line elements lying perpendicular to the middle surface of the plate remain
perpendicular to the middle surface during deformation, Fig. 6.1.6; this is similar
the “plane sections remain plane” assumption of the beam theory.
47. Assumptions of Plate Theory
(iii) Vertical strain is ignored
Line elements lying perpendicular to the mid-surface do not change length
during deformation, so that εzz = 0 throughout the plate. Again, this is similar
to an assumption of the beam theory.
These three assumptions are the basis of the Classical Plate Theory or the
Kirchhoff Plate Theory.
48. Buckling:
In structural engineering, buckling is the sudden change in shape (deformation) of a
structural component under load, such as the bowing of a column under compression or
the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing
load, when the load reaches a critical level, a member may suddenly change shape and
the structure and component is said to have buckled.
Buckling may occur even though the stresses that develop in the structure are well
below those needed to cause failure in the material of which the structure is composed.
Further loading may cause significant and somewhat unpredictable deformations,
possibly leading to complete loss of the member's load-carrying capacity.
However, if the deformations that occur after buckling do not cause the complete
collapse of that member, the member will continue to support the load that caused it to
buckle.
If the buckled member is part of a larger assemblage of components such as a building,
any load applied to the buckled part of the structure beyond that which caused the
member to buckle will be redistributed within the structure. Some aircraft are designed
for thin skin panels to continue carrying load even in the buckled state.
49. Buckling:
Most of steel or aluminum structures are made of tubes or welded plates.
Airplanes, ships and cars are assembled from metal plates pined by welling
riveting or spot welding.
Plated structures may fail by yielding fracture or buckling.
Buckling of thin plates occurs when a plate moves out of plane under
compressive load, causing it to bend in two directions.
The Bucking behavior of thin plates is significantly different from buckling
behavior of a column.
Buckling in a column terminates the members ability to resist axial force and as
a result , the critical load is the member’s failure load.
50. Buckling:
The same cannot be said for the buckling of thin plates due to the membrane
action of the plate
Plates under compression will continue to resist increasing axial force after
achieving the critical load , and will not fail until a load far greater than the
critical load is attained.
That shows that a plate’s critical load is not the same as its failure load .
51. Plastic Buckling:
When a material is loaded in compression it may buckle when a critical load is
applied.
If loading is performed at constant strain-rate, this initial buckling will be
elastic and will be recoverable when the applied compressive stress is reduced.
If loading is continued under these conditions, the buckled material may
deform enough to cause local plastic deformation to occur. This deformation is
permanent and cannot be recovered when the load is removed.
52. Example of Plastic Buckling :
The photograph shows a thin wall carbon-steel tube that has been buckled in
compression. The tube has a square section, and the plastic deformation is
self-constraining. Initially, the material deformed elastically. Upon reaching
the buckling threshold, it bowed out and plastic deformation was initiated at
the region of maximum curvature.
This "plastic hinge" can be folded at a lower applied stress than that needed
to initiate the buckle. When the material has closed on itself, a second hinge
is generated as the next tube section starts to buckle and plastically deform.
This process is repeated until the deformation is discontinued.