The document discusses bending, which refers to the behavior of a structural element subjected to an external load applied perpendicularly to its longitudinal axis. It describes Euler-Bernoulli beam theory, which assumes plane sections remain plane, and Timoshenko beam theory, which accounts for shear deformation. It also covers bending of plates, plastic bending, large bending deformations, and extensions of bending theories.
The document discusses bending of beams under transverse loads. It describes how bending causes stresses in beams, with compression on the top and tension on the bottom. It then provides the Euler-Bernoulli beam equation to model bending stresses as a function of the beam's properties and applied load. Finally, it discusses some extensions of the basic bending theory for non-uniform beams or large deformations.
This document discusses stresses in beams, specifically shear stresses. It covers five lectures on related topics like bending moment and shear force diagrams, bending stresses, shear stresses, deflection, and torsion. For shear stresses in beams with rectangular cross-sections, it explains that both normal and shear stresses are developed when loads produce both bending moments and shear forces. The maximum shear stress occurs at the center of the beam and its distribution is parabolic. Equations are provided for calculating shear stress values.
Unit 4 transverse loading on beams and stresses in beamskarthi keyan
This document discusses transverse loading on beams and stresses in beams. It defines a beam as a structural member used to bear different loads and resist vertical loads, shear forces, and bending moments. It describes different types of beams like cantilever beams and types of loads like point loads, uniformly distributed loads, and uniformly varying loads. It explains that shear force is the sum of forces on one side of a beam section, while bending moment is the sum of moments. It then discusses the theory of simple or pure bending, where a beam portion is only subjected to bending moment without shear force.
The document discusses various types of loading on structural members including pure bending, eccentric axial loading, and transverse loading. It covers bending deformations, strain and stress due to bending, section properties, and examples of bending stresses in composite and reinforced concrete beams. Plastic deformations in members made of elastic-plastic materials are also examined.
Vilas Nikam- Mechanics of Structure-Stress in beamNIKAMVN
1. The document discusses stresses in beams when subjected to external loading, specifically bending stress which is the resistance offered by internal stresses to bending.
2. It defines key concepts like neutral axis, section modulus, and presents flexural formulas for calculating bending stress based on moment of inertia, modulus of elasticity, and distance from neutral axis.
3. Various bending stress distributions are shown for different beam sections including rectangular, circular, hollow circular, and unsymmetrical sections for simply supported and cantilever beams. Shear stress distributions are also presented for several standard sections.
Like Comment and Download if u like this presentation
Motion and deformation of material under action of
Force
Temperature change
Phase change
Other external or internal agents
These changes lead us to some properties that are called Mechanical properties
Some of the Mechanical Properties
Ductility
Hardness
Impact resistance
Fracture toughness
Elasticity
Fatigue strength
Endurance limit
Creep resistance
Strength of material
Ductility: ductility is a solid material's ability to deform under tensile stress
Hardness of a material may refer to resistance to bending, scratching, abrasion or cutting.
Impact resistance is the ability of a material to withstand a high force or shock applied to it over a short period of time
Plasticity: ability of a material to deform permanently by the
This document discusses simple bending or pure bending of beams. It defines simple bending as when a length of a beam is subjected to zero shear force and constant bending moment. Under these conditions, the stresses introduced are called simple bending stresses. The key theories discussed include:
- The stress distribution under pure bending varies linearly with distance from the neutral axis.
- The neutral axis is the line where bending stresses are zero and coincides with the centroidal axis.
- The bending equation relates bending moment, flexural stress, moment of inertia and radius of curvature.
The document discusses bending of beams under transverse loads. It describes how bending causes stresses in beams, with compression on the top and tension on the bottom. It then provides the Euler-Bernoulli beam equation to model bending stresses as a function of the beam's properties and applied load. Finally, it discusses some extensions of the basic bending theory for non-uniform beams or large deformations.
This document discusses stresses in beams, specifically shear stresses. It covers five lectures on related topics like bending moment and shear force diagrams, bending stresses, shear stresses, deflection, and torsion. For shear stresses in beams with rectangular cross-sections, it explains that both normal and shear stresses are developed when loads produce both bending moments and shear forces. The maximum shear stress occurs at the center of the beam and its distribution is parabolic. Equations are provided for calculating shear stress values.
Unit 4 transverse loading on beams and stresses in beamskarthi keyan
This document discusses transverse loading on beams and stresses in beams. It defines a beam as a structural member used to bear different loads and resist vertical loads, shear forces, and bending moments. It describes different types of beams like cantilever beams and types of loads like point loads, uniformly distributed loads, and uniformly varying loads. It explains that shear force is the sum of forces on one side of a beam section, while bending moment is the sum of moments. It then discusses the theory of simple or pure bending, where a beam portion is only subjected to bending moment without shear force.
The document discusses various types of loading on structural members including pure bending, eccentric axial loading, and transverse loading. It covers bending deformations, strain and stress due to bending, section properties, and examples of bending stresses in composite and reinforced concrete beams. Plastic deformations in members made of elastic-plastic materials are also examined.
Vilas Nikam- Mechanics of Structure-Stress in beamNIKAMVN
1. The document discusses stresses in beams when subjected to external loading, specifically bending stress which is the resistance offered by internal stresses to bending.
2. It defines key concepts like neutral axis, section modulus, and presents flexural formulas for calculating bending stress based on moment of inertia, modulus of elasticity, and distance from neutral axis.
3. Various bending stress distributions are shown for different beam sections including rectangular, circular, hollow circular, and unsymmetrical sections for simply supported and cantilever beams. Shear stress distributions are also presented for several standard sections.
Like Comment and Download if u like this presentation
Motion and deformation of material under action of
Force
Temperature change
Phase change
Other external or internal agents
These changes lead us to some properties that are called Mechanical properties
Some of the Mechanical Properties
Ductility
Hardness
Impact resistance
Fracture toughness
Elasticity
Fatigue strength
Endurance limit
Creep resistance
Strength of material
Ductility: ductility is a solid material's ability to deform under tensile stress
Hardness of a material may refer to resistance to bending, scratching, abrasion or cutting.
Impact resistance is the ability of a material to withstand a high force or shock applied to it over a short period of time
Plasticity: ability of a material to deform permanently by the
This document discusses simple bending or pure bending of beams. It defines simple bending as when a length of a beam is subjected to zero shear force and constant bending moment. Under these conditions, the stresses introduced are called simple bending stresses. The key theories discussed include:
- The stress distribution under pure bending varies linearly with distance from the neutral axis.
- The neutral axis is the line where bending stresses are zero and coincides with the centroidal axis.
- The bending equation relates bending moment, flexural stress, moment of inertia and radius of curvature.
This document discusses bending stresses in beams. It defines simple or pure bending as when a beam is subjected to a bending moment that causes stresses but no shear stresses. The assumptions of pure bending theory are that the beam material is isotropic, homogeneous, initially straight, and elastic limits are not exceeded. Pure bending causes some layers to compress and others to tensile. A neutral axis experiences no stress. Bending stresses are calculated using the bending equation relating bending moment, moment of inertia, and distance from the neutral axis. Flitched or composite beams made of different materials also follow bending equations.
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.
A preparation for interview engineering mechanicsDr. Ramesh B
1. The document provides definitions and concepts related to engineering mechanics including: distinguishing between particles and rigid bodies, defining forces, moments, couples, friction, energy, momentum, and plane motion of rigid bodies.
2. Key concepts covered include the definition of a force, principle of resolution, Lami's theorem, resultant and equilibrant forces, conditions for equilibrium, types of beams and supports, types of loads, friction laws, forms of energy, impulse, momentum, types of impacts, and instantaneous centers of rotation.
3. Engineering mechanics concepts such as work-energy principle, conservation of energy, and the work-energy equation for general plane motion of rigid bodies are also stated.
This is a lecture on normal stress in mechanics of deformable bodies. There is a quick overview on what strength of materials is at the beginning of the presentation.
Presentation by:
MEC32/A1 Group 1 4Q 2014
MAGBOJOS, Redentor V.
RIGOR, Lady Krista V.
SALIDO, Lisette S.
Mapúa Institute of Technology
Presentation for Prof. Romeo D. Alastre's class.
This document provides an introduction to strength of materials (SOM). It defines key terms like strength, stiffness, stability, and durability. It discusses the basic problem in SOM as developing methods to design structural elements that consider strength, stiffness, stability, and economy. It also outlines the main hypotheses in SOM, including the material being continuous, homogeneous, and isotropic. It then discusses different types of stresses like tensile, compressive, and shear stresses. It provides stress-strain curves for ductile materials and defines modulus of elasticity. Examples of calculating stresses and strains in structural elements are also provided.
This document discusses different types of stresses that can occur in materials, including residual, structural, pressure, flow, thermal, and fatigue stresses. It also describes the basic types of internal stresses as tensile, compressive, and shear stresses. Tensile stress causes two parts of a material to pull apart, compressive stress causes two parts to press together, and shear stress causes two parts to slide across one another. The document lists various equipment used to test materials, such as universal testers, impact testers, hardness testers, and equipment for tensile, bend, compression, and flaring tests.
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.
Mechanics of materials deals with the relationship between external loads on a body and the internal loads within the body. It involves analyzing deformations and stability when subjected to forces. Equilibrium requires balancing all forces and moments on a body. Internal resultant loads include normal forces, shear forces, torques, and bending moments. Average normal stress is calculated as force over cross-sectional area. Average shear stress is calculated as shear force over cross-sectional area. A factor of safety is used to determine allowable loads based on failure loads to account for unknown factors.
This document discusses bending moment stress, including uniaxial and biaxial bending stress. It defines bending as causing deflection from a straight line, and bending moment as a measure of bending force applied at a distance from a beam's neutral axis. Bending moment stress induces tension or compression in a bent body. Examples show calculations for uniaxial 3-point and 4-point bending stresses, as well as an example of biaxial bending stress distribution and the line of zero stress in a loaded block.
This document contains lecture notes on mechanics of solids from the Department of Mechanical Engineering at Indus Institute of Technology & Engineering. It defines key concepts such as load, stress, strain, tensile stress and strain, compressive stress and strain, Young's modulus, shear stress and strain, shear modulus, stress-strain diagrams, working stress, and factor of safety. It also discusses thermal stresses, linear and lateral strain, Poisson's ratio, volumetric strain, bulk modulus, composite bars, bars with varying cross-sections, and stress concentration. The document provides examples to illustrate how to calculate stresses, strains, moduli, and other mechanical properties for different loading conditions.
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.
This study investigates the vibration characteristics of a cantilever beam made of linear elastic material with homogeneous and isotropic material properties. Static and modal analyses are performed to determine the stress, strain, deformation, natural frequencies, and mode shapes of the cantilever beam while it is being designed. The cantilever beam is modeled and analyzed in ANSYS to compare the stress and natural frequency for different materials with the same cross-sectional properties. The results show the deflection, stresses, and natural frequencies of the cantilever beam made of different materials.
The document discusses concepts related to stress analysis and design of structures including:
- Normal stress, shear stress, and bearing stress
- Stress analysis using statics to determine internal forces and stresses
- Design considerations like material selection and sizing based on allowable stresses
- Examples calculating stresses in rods, pins, and connections of a structure under a load.
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.
This document discusses different types of beams and loading conditions used in structural analysis. It defines dead load as the self-weight of building components and live load as external loads on a structure, which can be uniform, varying, or concentrated. Common beam types are described as simply supported, fixed, cantilever, continuous, and overhanging. Load types include concentrated, uniform distributed, uniformly varying, and applied couples. Shear force and bending moment are defined as the algebraic sum of vertical forces and moments acting on a beam cross section. Stress resultants in determinate beams can be calculated from equilibrium equations.
shear strain(due to shear stress&torsional stress)Mehedi Hasan
This document defines key concepts in mechanics of materials including stress, strain, shear stress, and torsion. It defines stress as the internal resistance to an externally applied load, and strain as the ratio of change in dimension to original dimension when a body is subjected to force. There are different types of strain including tensile, compressive, volumetric, and shear. Shear stress is defined similarly as the internal resistance to an applied shear load. The document also discusses Hooke's law and the linear relationship between shear stress and strain for elastic materials. Finally, it defines torsion as occurring when an external torque is applied, creating both an internal torque and shear stresses inside the material.
Analysis of complex composite beam by using timoshenko beam theoryIAEME Publication
This document summarizes an analysis of complex composite beams using Timoshenko beam theory and finite element methods. It begins with an abstract describing the motivation to analyze composite beams and lists some keywords. It then provides background on composite materials and discusses challenges in analyzing complex composite structures due to anisotropy. The document outlines the objectives, introduces Timoshenko beam theory, and describes the methodology used, which involves developing the governing equations, stiffness and mass matrices, and shape functions. It also summarizes the results and discussions, and lists references for further reading.
Flexural analysis of thick beams using singleiaemedu
This document presents a single variable shear deformation theory for flexural analysis of thick isotropic beams. The theory accounts for transverse shear deformation effects using a polynomial displacement field. The governing differential equation and boundary conditions are derived using the principle of virtual work. Results for displacement, stresses, and natural bending frequencies are obtained for simply supported thick beams under various loading cases and compared to exact solutions and other higher-order theories. The theory provides excellent accuracy for transverse shear stresses while avoiding the need for a shear correction factor.
1. The document analyzes the vibration of beams subjected to moving point loads using finite element analysis and the Newmark numerical time integration method.
2. It investigates the effect of load speed on the dynamic magnification factor, defined as the ratio of maximum dynamic displacement to static displacement.
3. The effect of spring stiffness at beam-column junctions is also evaluated. Computer codes in Matlab are developed to calculate dynamic responses and critical load velocities.
This document discusses bending stresses in beams. It defines simple or pure bending as when a beam is subjected to a bending moment that causes stresses but no shear stresses. The assumptions of pure bending theory are that the beam material is isotropic, homogeneous, initially straight, and elastic limits are not exceeded. Pure bending causes some layers to compress and others to tensile. A neutral axis experiences no stress. Bending stresses are calculated using the bending equation relating bending moment, moment of inertia, and distance from the neutral axis. Flitched or composite beams made of different materials also follow bending equations.
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.
A preparation for interview engineering mechanicsDr. Ramesh B
1. The document provides definitions and concepts related to engineering mechanics including: distinguishing between particles and rigid bodies, defining forces, moments, couples, friction, energy, momentum, and plane motion of rigid bodies.
2. Key concepts covered include the definition of a force, principle of resolution, Lami's theorem, resultant and equilibrant forces, conditions for equilibrium, types of beams and supports, types of loads, friction laws, forms of energy, impulse, momentum, types of impacts, and instantaneous centers of rotation.
3. Engineering mechanics concepts such as work-energy principle, conservation of energy, and the work-energy equation for general plane motion of rigid bodies are also stated.
This is a lecture on normal stress in mechanics of deformable bodies. There is a quick overview on what strength of materials is at the beginning of the presentation.
Presentation by:
MEC32/A1 Group 1 4Q 2014
MAGBOJOS, Redentor V.
RIGOR, Lady Krista V.
SALIDO, Lisette S.
Mapúa Institute of Technology
Presentation for Prof. Romeo D. Alastre's class.
This document provides an introduction to strength of materials (SOM). It defines key terms like strength, stiffness, stability, and durability. It discusses the basic problem in SOM as developing methods to design structural elements that consider strength, stiffness, stability, and economy. It also outlines the main hypotheses in SOM, including the material being continuous, homogeneous, and isotropic. It then discusses different types of stresses like tensile, compressive, and shear stresses. It provides stress-strain curves for ductile materials and defines modulus of elasticity. Examples of calculating stresses and strains in structural elements are also provided.
This document discusses different types of stresses that can occur in materials, including residual, structural, pressure, flow, thermal, and fatigue stresses. It also describes the basic types of internal stresses as tensile, compressive, and shear stresses. Tensile stress causes two parts of a material to pull apart, compressive stress causes two parts to press together, and shear stress causes two parts to slide across one another. The document lists various equipment used to test materials, such as universal testers, impact testers, hardness testers, and equipment for tensile, bend, compression, and flaring tests.
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.
Mechanics of materials deals with the relationship between external loads on a body and the internal loads within the body. It involves analyzing deformations and stability when subjected to forces. Equilibrium requires balancing all forces and moments on a body. Internal resultant loads include normal forces, shear forces, torques, and bending moments. Average normal stress is calculated as force over cross-sectional area. Average shear stress is calculated as shear force over cross-sectional area. A factor of safety is used to determine allowable loads based on failure loads to account for unknown factors.
This document discusses bending moment stress, including uniaxial and biaxial bending stress. It defines bending as causing deflection from a straight line, and bending moment as a measure of bending force applied at a distance from a beam's neutral axis. Bending moment stress induces tension or compression in a bent body. Examples show calculations for uniaxial 3-point and 4-point bending stresses, as well as an example of biaxial bending stress distribution and the line of zero stress in a loaded block.
This document contains lecture notes on mechanics of solids from the Department of Mechanical Engineering at Indus Institute of Technology & Engineering. It defines key concepts such as load, stress, strain, tensile stress and strain, compressive stress and strain, Young's modulus, shear stress and strain, shear modulus, stress-strain diagrams, working stress, and factor of safety. It also discusses thermal stresses, linear and lateral strain, Poisson's ratio, volumetric strain, bulk modulus, composite bars, bars with varying cross-sections, and stress concentration. The document provides examples to illustrate how to calculate stresses, strains, moduli, and other mechanical properties for different loading conditions.
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.
This study investigates the vibration characteristics of a cantilever beam made of linear elastic material with homogeneous and isotropic material properties. Static and modal analyses are performed to determine the stress, strain, deformation, natural frequencies, and mode shapes of the cantilever beam while it is being designed. The cantilever beam is modeled and analyzed in ANSYS to compare the stress and natural frequency for different materials with the same cross-sectional properties. The results show the deflection, stresses, and natural frequencies of the cantilever beam made of different materials.
The document discusses concepts related to stress analysis and design of structures including:
- Normal stress, shear stress, and bearing stress
- Stress analysis using statics to determine internal forces and stresses
- Design considerations like material selection and sizing based on allowable stresses
- Examples calculating stresses in rods, pins, and connections of a structure under a load.
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.
This document discusses different types of beams and loading conditions used in structural analysis. It defines dead load as the self-weight of building components and live load as external loads on a structure, which can be uniform, varying, or concentrated. Common beam types are described as simply supported, fixed, cantilever, continuous, and overhanging. Load types include concentrated, uniform distributed, uniformly varying, and applied couples. Shear force and bending moment are defined as the algebraic sum of vertical forces and moments acting on a beam cross section. Stress resultants in determinate beams can be calculated from equilibrium equations.
shear strain(due to shear stress&torsional stress)Mehedi Hasan
This document defines key concepts in mechanics of materials including stress, strain, shear stress, and torsion. It defines stress as the internal resistance to an externally applied load, and strain as the ratio of change in dimension to original dimension when a body is subjected to force. There are different types of strain including tensile, compressive, volumetric, and shear. Shear stress is defined similarly as the internal resistance to an applied shear load. The document also discusses Hooke's law and the linear relationship between shear stress and strain for elastic materials. Finally, it defines torsion as occurring when an external torque is applied, creating both an internal torque and shear stresses inside the material.
Analysis of complex composite beam by using timoshenko beam theoryIAEME Publication
This document summarizes an analysis of complex composite beams using Timoshenko beam theory and finite element methods. It begins with an abstract describing the motivation to analyze composite beams and lists some keywords. It then provides background on composite materials and discusses challenges in analyzing complex composite structures due to anisotropy. The document outlines the objectives, introduces Timoshenko beam theory, and describes the methodology used, which involves developing the governing equations, stiffness and mass matrices, and shape functions. It also summarizes the results and discussions, and lists references for further reading.
Flexural analysis of thick beams using singleiaemedu
This document presents a single variable shear deformation theory for flexural analysis of thick isotropic beams. The theory accounts for transverse shear deformation effects using a polynomial displacement field. The governing differential equation and boundary conditions are derived using the principle of virtual work. Results for displacement, stresses, and natural bending frequencies are obtained for simply supported thick beams under various loading cases and compared to exact solutions and other higher-order theories. The theory provides excellent accuracy for transverse shear stresses while avoiding the need for a shear correction factor.
1. The document analyzes the vibration of beams subjected to moving point loads using finite element analysis and the Newmark numerical time integration method.
2. It investigates the effect of load speed on the dynamic magnification factor, defined as the ratio of maximum dynamic displacement to static displacement.
3. The effect of spring stiffness at beam-column junctions is also evaluated. Computer codes in Matlab are developed to calculate dynamic responses and critical load velocities.
1) Euler-Bernoulli bending theory and Timoshenko beam theory describe the stresses and deflections of beams under bending loads.
2) Euler-Bernoulli theory assumes a beam's cross-section remains plane and perpendicular to the neutral axis during bending. Timoshenko theory accounts for shear deformation.
3) Both theories relate the bending moment M and shear force V to the beam's deflection w and its derivatives, allowing calculation of stresses, forces, and deflections for given beam geometries and loads.
The use of Calculus is very important in every aspects of engineering.
The use of Differential equation is very much applied in the concept of Elastic beams.
This document describes the design and fabrication of a shaft drive for a bicycle that transmits power from the pedals to the rear wheel using a drive shaft instead of a chain. It discusses how bevel gears are used to change the direction of rotation from the pedals to the perpendicular direction of the rear wheel. By avoiding the use of a chain and sprockets, friction and wear are reduced. The document provides background on the history of shaft-driven bicycles and discusses their advantages over chain-driven designs, including greater efficiency. It also reviews different types of drive shafts used in automobiles and their functions in transmitting torque from the engine to the rear wheels.
The document discusses the history and importance of chocolate in human civilization. It notes that chocolate originated in Mesoamerica over 3000 years ago and was prized by the Aztecs and Mayans for its taste. Cocoa beans were used as currency and their cultivation was tightly regulated. The Spanish brought cocoa to Europe in the 16th century, starting its global spread and the development of the chocolate industry.
Module 4 flexural stresses- theory of bendingAkash Bharti
This document provides an overview of flexural stresses and the theory of simple bending. It discusses key concepts such as:
- Assumptions in the derivation of the bending equation relating bending moment (M) to curvature (1/R) and stress (f)
- Determining the neutral axis where bending stress is zero
- Calculating bending stresses in beams undergoing simple bending and pure bending
- Deriving Bernoulli's bending equation relating stress (f) to distance from the neutral axis (y) and bending moment (M)
- Using the bending equation to locate the neutral axis and design beam cross-sections based on permissible stresses
Worked examples are provided to illustrate calculating load capacity based on beam geometry and material properties
This document discusses stresses, strains, and mechanical properties of materials as they relate to biomechanics and bone behavior. It contains the following key points:
1. Stresses are defined as forces per unit area and strains are defined as deformations or changes in length. Normal stresses act perpendicular to a plane, while shear stresses act parallel.
2. Materials experience different types of stresses and strains including compression, tension, bending, torsion, and shear. Stress-strain curves are used to characterize elastic behavior, yield points, and failure properties.
3. Bone is a non-homogeneous, anisotropic, viscoelastic composite material that can withstand different loading rates and stresses in longitudinal and transverse directions. Its
The document provides information about aircraft structures and structural analysis. It defines primary and secondary aircraft structures, and monocoque and semi-monocoque constructions. It describes the functions of different structural elements like the skin, ribs, spars, and stringers. It discusses simplifications made in structural analysis, including lumping longitudinal members and assuming webs only experience shear stresses. It also covers topics like unsymmetric bending, shear flow, shear centers, and torsion of thin-walled closed sections.
This document discusses the mechanical properties of solids, including elasticity, plasticity, stress, strain, elastic limit, Hooke's law, modulus of elasticity, and stress-strain curves. It defines key terms and concepts related to how solids deform under force. Examples are given of how understanding mechanical properties informs applications like designing ropes for cranes and bridges to withstand loads within safe elastic limits. The maximum possible height of mountains is also calculated based on the shear modulus of typical rock.
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.
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 discusses torsional stress and its effects on beams and circular shafts. It makes the following key points:
1. Torsional stress is the shear stress developed in a material subjected to a twisting torque, and is highest at the outermost parts of the material furthest from the central axis.
2. Cracks form under torsional stress and initially appear at the middle of the longest side of a beam, then the shortest side, before circulating around the beam's periphery.
3. Failure of brittle materials under pure torsion occurs along planes inclined to the beam axis, not perpendicular, due to elastic theory principles.
Whenever a body is subjected to an axial tension or compression, a direct stress comes into play at every section of body. We also know that whenever a body is subjected to a bending moment a bending moment a bending stress comes into play.
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 first chapter explains the engineering fundamentals of stress analysis.
I. The course aims to enable students to relate material properties to behavior under loads, analyze loaded structural members, and evaluate stresses, strains, and deflections.
II. The course structure covers stresses and strains, shear force and bending moment diagrams, flexural and shear stresses in beams, torsion of circular shafts, and columns and struts.
III. Teaching methods include lectures involving tutorial solutions, coursework assignments, and daily assessment. The course examines topics like stress-strain relationships, thermal and volumetric strains, Hooke's law, modulus of elasticity, yield stresses, and factors of safety.
This document discusses bending moments and shear forces in beams. It defines different types of beams such as simply supported beams, cantilever beams, and beams with overhangs. It also defines types of loads like concentrated loads, distributed loads, and couples. It explains how to calculate the shear force and bending moment at any cross-section of a beam and discusses relationships between loads, shear forces and bending moments. It provides examples of drawing shear force and bending moment diagrams. Finally, it discusses bending stresses in beams and bending of beams made of two materials.
B Ending Moments And Shearing Forces In Beams2Amr Hamed
This document discusses bending moments and shear forces in beams. It defines different types of beams such as simply supported beams, cantilever beams, and beams with overhangs. It also defines types of loads like concentrated loads, distributed loads, and couples. It explains how to calculate the shear force and bending moment at any cross-section of a beam and discusses relationships between loads, shear forces and bending moments. It provides examples of drawing shear force and bending moment diagrams. Finally, it discusses bending stresses in beams and bending of beams made of two materials.
This document discusses bending moments and shear forces in beams. It defines different types of beams such as simply supported beams, cantilever beams, and beams with overhangs. It also defines types of loads like concentrated loads, distributed loads, and couples. It explains how to calculate the shear force and bending moment at any cross-section of a beam and discusses relationships between loads, shear forces and bending moments. It provides examples of drawing shear force and bending moment diagrams. Finally, it discusses bending stresses in beams and bending of beams made of two materials.
B Ending Moments And Shearing Forces In Beams2Amr Hamed
This document discusses bending moments and shear forces in beams. It defines different types of beams such as simply supported beams, cantilever beams, and beams with overhangs. It also defines types of loads like concentrated loads, distributed loads, and couples. It explains how to calculate the shear force and bending moment at any cross-section of a beam and discusses relationships between loads, shear forces and bending moments. It provides examples of drawing shear force and bending moment diagrams. Finally, it discusses bending stresses in beams and bending of beams made of two materials.
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This document provides a summary of key concepts in strength of materials for mechanical engineers. It defines terms like stress, strain, Hooke's law, moment of force, couple, center of gravity, moment of inertia, shear stress, Poisson's ratio, bulk modulus, principal plane and stress, Mohr's circle, resilience, malleability and ductility. It also discusses different types of beams, loading, shear force, bending moment, riveted joints, pitch and margin. The document aims to give a quick brush up on important topics in strength of materials through concise definitions and explanations of key terms and concepts.
This document discusses stress-strain curves and various material testing methods. It contains the following key points:
1. Creep testing involves applying a constant load to a material sample at high temperature and measuring deformation over time to evaluate materials performance. Fatigue testing subjects samples to repeated stresses to determine fatigue strength.
2. Stress-strain curves relate the stress and strain experienced by materials. They contain useful data like proportional limit, elastic limit, yield point, ultimate strength, and ductile vs. brittle fracture behavior.
3. True stress-strain diagrams account for changes in cross-sectional area during testing, while engineering stress-strain curves do not. Both are commonly used in design as long as strains remain
This document discusses stress-strain curves and various material testing methods. It contains the following key points:
1. Creep testing involves applying a constant load to a material sample at high temperature and measuring deformation over time to evaluate materials performance. Fatigue testing subjects samples to repeated stresses to determine fatigue strength.
2. Stress-strain curves relate the stress and strain experienced by materials. They contain useful data like proportional limit, elastic limit, yield point, ultimate strength, and ductile vs. brittle fracture behavior.
3. True stress-strain curves account for changes in cross-sectional area during testing, providing a more accurate representation of material behavior, though engineering stress-strain curves are sufficient for most design
This document discusses stress-strain curves for different materials. It explains that ductile materials like steel exhibit a linear elastic region followed by a plastic region and ultimate failure. The yield point marks the transition from elastic to plastic deformation. Brittle materials like concrete do not have a yield point and fail elastically without plastic deformation. Poisson's ratio describes the contraction of a material perpendicular to the direction of stretching or expansion. It is an important material property used in applications like piping where internal pressure causes radial and axial deformation. Temperature changes also induce thermal stresses and strains as materials expand at higher temperatures.
1. The intertestamental period refers to the approximately 400 years between the events of the Old Testament and the appearance of Jesus Christ.
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Wood is a hard, fibrous material found in trees and woody plants. It is composed of cellulose fibers embedded in a matrix of lignin. Wood has been used for thousands of years as a construction material and fuel source. The earth contains about one trillion tonnes of wood, which grows at a rate of 10 billion tonnes per year. In 1991, approximately 3.5 billion cubic meters of wood were harvested globally, primarily for furniture and building construction.
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1) A cent is a unit of measurement that is equal to 1/100 of another unit. In India, 1 cent equals 1/100 of an acre, which is equal to 40.468 square meters.
2) Calculations show that 100 cents equals 1 acre, 1 cent equals 1/100 acre or 40.467 square meters. Conversions between cents, acres, and other area units like square feet and square yards are also provided.
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Brickwork is the construction of walls and structures using bricks laid in mortar. Some key points:
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- Different brick bonds like Flemish bond arrange stretchers and headers in patterns for strength.
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A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Bending wikipedia
1. 9/6/13 Bending - Wikipedia, the free encyclopedia
en.wikipedia.org/wiki/Bending 1/12
Bending of an I-beam
Bending
From Wikipedia, the free encyclopedia
In engineering mechanics, bending (also known as flexure)
characterizes the behavior of a slender structural element
subjected to an external load applied perpendicularly to a
longitudinal axis of the element.
The structural element is assumed to be such that at least one
of its dimensions is a small fraction, typically 1/10 or less, of the
other two.[1] When the length is considerably longer than the
width and the thickness, the element is called a beam. For
example, a closet rod sagging under the weight of clothes on
clothes hangers is an example of a beam experiencing bending.
On the other hand, a shell is a structure of any geometric form
where the length and the width are of the same order of magnitude but the thickness of the structure (known as the
'wall') is considerably smaller. A large diameter, but thin-walled, short tube supported at its ends and loaded
laterally is an example of a shell experiencing bending.
In the absence of a qualifier, the term bending is ambiguous because bending can occur locally in all objects. To
make the usage of the term more precise, engineers refer to the bending of rods,[2] the bending of beams,[1] the
bending of plates,[3] the bending of shells[2] and so on.
Contents
1 Quasistatic bending of beams
1.1 Euler-Bernoulli bending theory
1.2 Extensions of Euler-Bernoulli beam bending theory
1.2.1 Plastic bending
1.2.2 Complex or asymmetrical bending
1.2.3 Large bending deformation
1.3 Timoshenko bending theory
2 Dynamic bending of beams
2.1 Euler-Bernoulli theory
2.1.1 Free vibrations
2.2 Timoshenko-Rayleigh theory
2.2.1 Free vibrations
3 Quasistatic bending of plates
3.1 Kirchhoff-Love theory of plates
3.2 Mindlin-Reissner theory of plates
4 Dynamic bending of plates
4.1 Dynamics of thin Kirchhoff plates
5 See also
6 References
2. 9/6/13 Bending - Wikipedia, the free encyclopedia
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Element of a bent beam: the fibers form concentric arcs, the
top fibers are compressed and bottom fibers stretched.
7 External links
Quasistatic bending of beams
A beam deforms and stresses develop inside it when a transverse load is applied on it. In the quasistatic case, the
amount of bending deflection and the stresses that develop are assumed not to change over time. In a horizontal
beam supported at the ends and loaded downwards in the middle, the material at the over-side of the beam is
compressed while the material at the underside is stretched. There are two forms of internal stresses caused by
lateral loads:
Shear stress parallel to the lateral loading plus complementary shear stress on planes perpendicular to the
load direction;
Direct compressive stress in the upper region of the beam, and direct tensile stress in the lower region of the
beam.
These last two forces form a couple or moment as they are equal in magnitude and opposite in direction. This
bending moment resists the sagging deformation characteristic of a beam experiencing bending. The stress
distribution in a beam can be predicted quite accurately even when some simplifying assumptions are used.[1]
Euler-Bernoulli bending theory
Main article: Euler-Bernoulli beam equation
In the Euler-Bernoulli theory of slender beams, a
major assumption is that 'plane sections remain
plane'. In other words, any deformation due to
shear across the section is not accounted for (no
shear deformation). Also, this linear distribution is
only applicable if the maximum stress is less than
the yield stress of the material. For stresses that
exceed yield, refer to article plastic bending. At
yield, the maximum stress experienced in the
section (at the furthest points from the neutral axis
of the beam) is defined as the flexural strength.
The Euler-Bernoulli equation for the quasistatic
bending of slender, isotropic, homogeneous
beams of constant cross-section under an applied
transverse load is[1]
where is the Young's modulus, is the area moment of inertia of the cross-section, and is the deflection
of the neutral axis of the beam.
3. 9/6/13 Bending - Wikipedia, the free encyclopedia
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Bending moments in a beam
After a solution for the displacement of the beam has been obtained, the bending moment ( ) and shear force (
) in the beam can be calculated using the relations
Simple beam bending is often analyzed with the Euler-Bernoulli beam equation. The conditions for using simple
bending theory are:[4]
1. The beam is subject to pure bending. This
means that the shear force is zero, and that
no torsional or axial loads are present.
2. The material is isotropic and homogeneous.
3. The material obeys Hooke's law (it is
linearly elastic and will not deform
plastically).
4. The beam is initially straight with a cross
section that is constant throughout the
beam length.
5. The beam has an axis of symmetry in the
plane of bending.
6. The proportions of the beam are such that
it would fail by bending rather than by crushing, wrinkling or sideways buckling.
7. Cross-sections of the beam remain plane during bending.
Compressive and tensile forces develop in the direction of the beam axis under bending loads. These forces induce
stresses on the beam. The maximum compressive stress is found at the uppermost edge of the beam while the
maximum tensile stress is located at the lower edge of the beam. Since the stresses between these two opposing
maxima vary linearly, there therefore exists a point on the linear path between them where there is no bending
stress. The locus of these points is the neutral axis. Because of this area with no stress and the adjacent areas with
low stress, using uniform cross section beams in bending is not a particularly efficient means of supporting a load as
it does not use the full capacity of the beam until it is on the brink of collapse. Wide-flange beams (I-beams) and
truss girders effectively address this inefficiency as they minimize the amount of material in this under-stressed
region.
The classic formula for determining the bending stress in a beam under simple bending is:[5]
where
is the bending stress
M - the moment about the neutral axis
y - the perpendicular distance to the neutral axis
Ix - the second moment of area about the neutral axis x.
4. 9/6/13 Bending - Wikipedia, the free encyclopedia
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Extensions of Euler-Bernoulli beam bending theory
Plastic bending
Main article: Plastic bending
The equation is valid only when the stress at the extreme fiber (i.e., the portion of the beam farthest from
the neutral axis) is below the yield stress of the material from which it is constructed. At higher loadings the stress
distribution becomes non-linear, and ductile materials will eventually enter a plastic hinge state where the
magnitude of the stress is equal to the yield stress everywhere in the beam, with a discontinuity at the neutral axis
where the stress changes from tensile to compressive. This plastic hinge state is typically used as a limit state in the
design of steel structures.
Complex or asymmetrical bending
The equation above is only valid if the cross-section is symmetrical. For homogeneous beams with asymmetrical
sections, the axial stress in the beam is given by
[6]
where are the coordinates of a point
on the cross section at which the stress is
to be determined as shown to the right,
and are the bending moments
about the y and z centroid axes, and
are the second moments of area (distinct
from moments of inertia) about the y and z
axes, and is the product of moments of
area. Using this equation it is possible to
calculate the bending stress at any point on
the beam cross section regardless of
moment orientation or cross-sectional
shape. Note that
do not change from one point to another
on the cross section.
Large bending deformation
For large deformations of the body, the
stress in the cross-section is calculated
using an extended version of this formula.
First the following assumptions must be
made:
1. Assumption of flat sections - before
5. 9/6/13 Bending - Wikipedia, the free encyclopedia
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Deflection of a beam deflected symmetrically and principle of
superposition
and after deformation the
considered section of body remains
flat (i.e., is not swirled).
2. Shear and normal stresses in this
section that are perpendicular to the
normal vector of cross section have no
influence on normal stresses that are parallel to
this section.
Large bending considerations should be implemented
when the bending radius is smaller than ten section
heights h:
With those assumptions the stress in large bending is calculated as:
where
is the normal force
is the section area
is the bending moment
is the local bending radius (the radius of bending at the current section)
is the area moment of inertia along the x-axis, at the place (see Steiner's theorem)
is the position along y-axis on the section area in which the stress is calculated.
When bending radius approaches infinity and , the original formula is back:
.
Timoshenko bending theory
Main article: Timoshenko beam theory
In 1921, Timoshenko improved upon the Euler-Bernoulli theory of beams by adding the effect of shear into the
beam equation. The kinematic assumptions of the Timoshenko theory are:
normals to the axis of the beam remain straight after deformation
there is no change in beam thickness after deformation
However, normals to the axis are not required to remain perpendicular to the axis after deformation.
The equation for the quasistatic bending of a linear elastic, isotropic, homogeneous beam of constant cross-section
beam under these assumptions is[7]
6. 9/6/13 Bending - Wikipedia, the free encyclopedia
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Deformation of a Timoshenko
beam. The normal rotates by an
amount which is not equal to
.
where is the area moment of inertia of the cross-section, is the cross-sectional area, is the shear modulus,
and is a shear correction factor. For materials with Poisson's ratios ( ) close to 0.3, the shear correction factor
for a rectangular cross-section is approximately
The rotation ( ) of the normal is described by the equation
The bending moment ( ) and the shear force ( ) are given by
Dynamic bending of beams
The dynamic bending of beams,[8] also known as flexural vibrations of beams, was first investigated by Daniel
Bernoulli in the late 18th century. Bernoulli's equation of motion of a vibrating beam tended to overestimate the
natural frequencies of beams and was improved marginally by Rayleigh in 1877 by the addition of a mid-plane
rotation. In 1921 Stephen Timoshenko improved the theory further by incorporating the effect of shear on the
dynamic response of bending beams. This allowed the theory to be used for problems involving high frequencies of
vibration where the dynamic Euler-Bernoulli theory is inadequate. The Euler-Bernoulli and Timoshenko theories for
the dynamic bending of beams continue to be used widely by engineers.
Euler-Bernoulli theory
Main article: Euler-Bernoulli beam equation
The Euler-Bernoulli equation for the dynamic bending of slender, isotropic, homogeneous beams of constant cross-
section under an applied transverse load is[7]
7. 9/6/13 Bending - Wikipedia, the free encyclopedia
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where is the Young's modulus, is the area moment of inertia of the cross-section, is the deflection of
the neutral axis of the beam, and is mass per unit length of the beam.
Free vibrations
For the situation where there is no transverse load on the beam, the bending equation takes the form
Free, harmonic vibrations of the beam can then be expressed as
and the bending equation can be written as
The general solution of the above equation is
where are constants and
8. 9/6/13 Bending - Wikipedia, the free encyclopedia
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The mode shapes of a cantilevered I-beam
1st lateral bending 1st torsional 1st vertical bending
2nd lateral bending 2nd torsional 2nd vertical bending
Timoshenko-Rayleigh theory
Main article: Timoshenko beam theory
In 1877, Rayleigh proposed an improvement to the dynamic Euler-Bernoulli beam theory by including the effect of
rotational inertia of the cross-section of the beam. Timoshenko improved upon that theory in 1922 by adding the
effect of shear into the beam equation. Shear deformations of the normal to the mid-surface of the beam are
allowed in the Timoshenko-Rayleigh theory.
The equation for the bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam under
these assumptions is [7][9]
where is the polar moment of inertia of the cross-section, is the mass per unit length of the beam, is the density
of the beam, is the cross-sectional area, is the shear modulus, and is a shear correction factor. For
materials with Poisson's ratios ( ) close to 0.3, the shear correction factor are approximately
Free vibrations
For free, harmonic vibrations the Timoshenko-Rayleigh equations take the form
9. 9/6/13 Bending - Wikipedia, the free encyclopedia
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Deformation of a thin plate highlighting the
displacement, the mid-surface (red) and
the normal to the mid-surface (blue)
This equation can be solved by noting that all the derivatives of must have the same form to cancel out and hence
as solution of the form may be expected. This observation leads to the characteristic equation
The solutions of this quartic equation are
where
The general solution of the Timoshenko-Rayleigh beam equation for free vibrations can then be written as
Quasistatic bending of plates
Main article: Plate theory
The defining feature of beams is that one of the dimensions is much
larger than the other two. A structure is called a plate when it is flat
and one of its dimensions is much smaller than the other two. There
are several theories that attempt to describe the deformation and
stress in a plate under applied loads two of which have been used
widely. These are
the Kirchhoff-Love theory of plates (also called classical
plate theory)
the Mindlin-Reissner plate theory (also called the first-order
shear theory of plates)
Kirchhoff-Love theory of plates
The assumptions of Kirchhoff-Love theory are
straight lines normal to the mid-surface remain straight after
deformation
straight lines normal to the mid-surface remain normal to the
mid-surface after deformation
the thickness of the plate does not change during a deformation.
These assumptions imply that
where is the displacement of a point in the plate and is the displacement of the mid-surface.
10. 9/6/13 Bending - Wikipedia, the free encyclopedia
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The strain-displacement relations are
The equilibrium equations are
where is an applied load normal to the surface of the plate.
In terms of displacements, the equilibrium equations for an isotropic, linear elastic plate in the absence of external
load can be written as
In direct tensor notation,
Mindlin-Reissner theory of plates
The special assumption of this theory is that normals to the mid-surface remain straight and inextensible but not
necessarily normal to the mid-surface after deformation. The displacements of the plate are given by
where are the rotations of the normal.
The strain-displacement relations that result from these assumptions are
where is a shear correction factor.
The equilibrium equations are
where
Dynamic bending of plates
Main article: Plate theory
Dynamics of thin Kirchhoff plates
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The dynamic theory of plates determines the propagation of waves in the plates, and the study of standing waves
and vibration modes. The equations that govern the dynamic bending of Kirchhoff plates are
where, for a plate with density ,
and
The figures below show some vibrational modes of a circular plate.
mode k = 0, p = 1 mode k = 0, p = 2
mode k = 1, p = 2
See also
Bending moment
Bending Machine (flat metal bending)
Brake (sheet metal bending)
Bending of plates
Bending (metalworking)
Contraflexure
Flexure bearing
List of area moments of inertia
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Shear and moment diagram
Shear strength
Sandwich theory
Vibration
Vibration of plates
Brazier effect
References
1. ^ a b c d Boresi, A. P. and Schmidt, R. J. and Sidebottom, O. M., 1993, Advanced mechanics of materials, John
Wiley and Sons, New York.
2. ^ a b Libai, A. and Simmonds, J. G., 1998, The nonlinear theory of elastic shells, Cambridge University Press.
3. ^ Timoshenko, S. and Woinowsky-Krieger, S., 1959, Theory of plates and shells, McGraw-Hill.
4. ^ Shigley J, "Mechanical Engineering Design", p44, International Edition, pub McGraw Hill, 1986, ISBN 0-07-
100292-8
5. ^ Gere, J. M. and Timoshenko, S.P., 1997, Mechanics of Materials, PWS Publishing Company.
6. ^ Cook and Young, 1995, Advanced Mechanics of Materials, Macmillan Publishing Company: New York
7. ^ a b c Thomson, W. T., 1981, Theory of Vibration with Applications
8. ^ Han, S. M, Benaroya, H. and Wei, T., 1999, "Dynamics of transversely vibrating beams using four engineering
theories," Journal of Sound and Vibration, vol. 226, no. 5, pp. 935-988.
9. ^ Rosinger, H. E. and Ritchie, I. G., 1977, On Timoshenko's correction for shear in vibrating isotropic beams, J.
Phys. D: Appl. Phys., vol. 10, pp. 1461-1466.
External links
Flexure formulae (http://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/flexure-formula)
Beam flexure, stress formulae and calculators (http://www.engineersedge.com/beam_calc_menu.shtml)
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Categories: Statics Elasticity (physics) Solid mechanics Structural system Deformation
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