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 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.
Introduction-Plastic hinge concept-plastic section modulus-shape factor-redistribution of moments-collapse mechanism.
Theorems of plastic analysis - Static/lower bound theorem; Kinematic/upper bound theorem-Plastic analysis of beams and portal frames by equilibrium and mechanism methods.
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 influence lines for beams. Influence lines graphically show how a unit load moving across a structure affects structural responses like reactions, shears, moments, and deflections. The ordinates of an influence line diagram represent the magnitude and type of a structural response when a unit load is at that point. Influence lines are useful for analyzing the effects of moving loads on structures. They can be used to determine locations of loads that cause maximum structural responses, which can then be calculated. The document also provides properties of influence lines and discusses how to qualitatively and quantitatively draw them for beams.
This document discusses beam design criteria and deflection behavior of beams. It covers two key criteria for beam design:
1) Strength criterion - the beam cross section must be strong enough to resist bending moments and shear forces.
2) Stiffness criterion - the maximum deflection of the beam cannot exceed a limit and the beam must be stiff enough to resist deflections from loading.
It then defines deflection, slope, elastic curve, and flexural rigidity. It presents the differential equation that relates bending moment, slope, and deflection. Methods for calculating slope and deflection including double integration, Macaulay's method, and others are also summarized.
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.
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.
Introduction-Plastic hinge concept-plastic section modulus-shape factor-redistribution of moments-collapse mechanism.
Theorems of plastic analysis - Static/lower bound theorem; Kinematic/upper bound theorem-Plastic analysis of beams and portal frames by equilibrium and mechanism methods.
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 influence lines for beams. Influence lines graphically show how a unit load moving across a structure affects structural responses like reactions, shears, moments, and deflections. The ordinates of an influence line diagram represent the magnitude and type of a structural response when a unit load is at that point. Influence lines are useful for analyzing the effects of moving loads on structures. They can be used to determine locations of loads that cause maximum structural responses, which can then be calculated. The document also provides properties of influence lines and discusses how to qualitatively and quantitatively draw them for beams.
This document discusses beam design criteria and deflection behavior of beams. It covers two key criteria for beam design:
1) Strength criterion - the beam cross section must be strong enough to resist bending moments and shear forces.
2) Stiffness criterion - the maximum deflection of the beam cannot exceed a limit and the beam must be stiff enough to resist deflections from loading.
It then defines deflection, slope, elastic curve, and flexural rigidity. It presents the differential equation that relates bending moment, slope, and deflection. Methods for calculating slope and deflection including double integration, Macaulay's method, and others are also summarized.
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 discusses the shear center of beam sections. It defines the shear center as the point where a load can be applied to cause pure bending without any twisting. It then provides properties of the shear center, including that it lies on the axis of symmetry for some sections. Methods for determining the location of the shear center are presented, including using the first moment of area. Real-life examples of applying shear center concepts to purlins and channel sections are given. The document concludes with an example problem of locating the shear center and calculating shear stresses for a hat section.
Relation between load shear force and bending moment of beamssushma chinta
This document discusses the relationships between loads, shear forces, and bending moments in beams. It states that shear forces and bending moments are internal stress resultants that can be calculated from equations of equilibrium. Distributed loads cause shear forces to vary linearly or quadratically along the beam and bending moments to vary quadratically or cubically. Concentrated loads cause an abrupt change in shear force but no change in bending moment. Couples cause no change in shear force but an abrupt change in bending moment.
Structural Integrity Analysis: Chapter 3 Mechanical Properties of MaterialsIgor Kokcharov
Structural Integrity Analysis features a collection of selected topics on structural design, safety, reliability, redundancy, strength, material science, mechanical properties of materials, composite materials, welds, finite element analysis, stress concentration, failure mechanisms and criteria. The engineering approaches focus on understanding and concept visualization rather than theoretical reasoning. The structural engineering profession plays a key role in the assurance of safety of technical systems such as metallic structures, buildings, machines, and transport. The third chapter explains the engineering tests and fundamentals of mechanical properties of materials.
This document summarizes concepts related to torsion and the torsion of circular elastic bars. It discusses the assumptions made in analyzing torsion, including that shear strain varies linearly from the central axis. It also covers determining shear stress and torque using the polar moment of inertia for circular cross-sections. The relationships between applied torque, shear stress, shear strain, and angle of twist are defined. Stress concentrations and alternative differential equations approaches are also summarized.
This document discusses statically indeterminate structures and thermal stresses. It begins by defining statically indeterminate structures as those where the number of unknowns is greater than the number of equilibrium equations, requiring additional equations. It provides examples of compound bars made of two materials, where the deformations are equal and stresses can be calculated. It also discusses temperature stresses that develop when a material is prevented from expanding or contracting freely due to a temperature change. The temperature strain and stress formulas are provided. Several example problems are then solved to calculate stresses, deformations and loads for statically indeterminate structures and those subjected to temperature changes.
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.
This document describes different types of beams based on their end support, cross-section shape, equilibrium condition, and geometry. Beams can be simply supported, continuous, overhanging, cantilever, fixed, or trussed based on their end support. Their cross-section can be I-beams, T-beams, or C-beams. Based on equilibrium, beams are either statically determinate or indeterminate. A beam's geometry can be straight, curved, or tapered.
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 provides an overview of reinforced concrete design principles for civil engineers and construction managers. It discusses the aim of structural design according to BS 8110, describes the properties and composite action of reinforced concrete, explains limit state design methodology, and summarizes key elements like slabs, beams, columns, walls, and foundations. The document also covers material properties, stress-strain curves, failure modes, and general procedures for slab sizing and design.
Pure bending of curved bar (polar coordinate)Pratish Sardar
This document discusses pure bending of curved bars. It presents the assumptions made in analyzing curved members under bending, including that plane cross sections remain plane after bending. It defines key parameters like the radii of the inner and outer fibers. The stress function is introduced and the equations for radial and tangential stresses are given. The boundary conditions for a curved bar under pure bending are described. Standard relations are presented for determining the coefficients of the stress function based on the boundary conditions. Radial and circumferential stress equations are provided in terms of these coefficients. Finally, the document indicates it will provide numerical examples.
So far, all of the exercises presented in this module have been statically determinate, i.e. there have been enough equations of equilibrium available to solve for the unknowns. This final section will be concerned with statically indeterminate structures, and two methods used to solve these problems will be presented.
This document provides an introduction to strength of materials, including concepts of stress, strain, Hooke's law, stress-strain relationships, elastic constants, and factors of safety. It defines key terms like stress, strain, elastic limit, modulus of elasticity, and ductile and brittle material behavior. Examples of stress and strain calculations are provided for basic structural elements like rods, bars, and composite structures. The document also covers compound bars, principle of superposition, and effects of temperature changes.
This document discusses beam deflection. It begins by defining beam deflection and the factors that affect it, including bending moment, material properties, and shape properties. It then presents the general formula for calculating beam deflection using double integration of the bending moment equation. Examples are given of using boundary conditions to solve for deflection in simply supported beams, cantilever beams, and beams under various loading types. Common deflection formulas are also presented.
This document summarizes key concepts related to structural analysis including:
1) The effects of axial and eccentric loading on columns including direct stress, bending stress, and maximum/minimum stresses.
2) Maximum and minimum pressures at the base of dams and retaining walls including calculations of total water/earth pressure, eccentricity, and stability conditions.
3) Forces and stresses on chimneys and walls due to wind pressure including calculations of direct stress from self-weight, wind force, induced bending moment, and maximum/minimum stresses.
The document discusses bending stresses in beams. It describes how bending stresses are developed in beams to resist bending moments and shearing forces. The theory of pure bending is introduced, where only bending stresses are considered without the effect of shear. Equations for calculating bending stresses are derived based on the beam's moment of inertia, bending moment, and distance from the neutral axis. Several beam cross-section examples are provided to demonstrate how to calculate the maximum bending stress and section modulus.
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.
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 beam theory and provides equations for calculating the deflection and slope of beams under different loading conditions. It defines key terms like bending moment, radius of curvature, flexural stiffness, and provides equations relating these terms. Specifically, it gives the deflection and slope equations for a cantilever beam with a point load, cantilever with uniform load, simply supported beam with central point load, and simply supported beam with uniform load.
we select cantilever beam having I,C,T section and we select material cast iron, stainless steel, steel and analyze base upon modal and static analysis.we see here deformation,stress ,strain and based upon it we conclude.
This document provides information about beams used in structural engineering. It defines beams, discusses their structural characteristics like moment of inertia and stresses, and describes different types of beams including simply supported, fixed, cantilever, and trussed beams. It also covers beam design, applications in bridges and cranes, potential failure modes from plastic hinges, buckling or material failure, and methods to prevent failures like lateral restraints.
This document discusses the shear center of beam sections. It defines the shear center as the point where a load can be applied to cause pure bending without any twisting. It then provides properties of the shear center, including that it lies on the axis of symmetry for some sections. Methods for determining the location of the shear center are presented, including using the first moment of area. Real-life examples of applying shear center concepts to purlins and channel sections are given. The document concludes with an example problem of locating the shear center and calculating shear stresses for a hat section.
Relation between load shear force and bending moment of beamssushma chinta
This document discusses the relationships between loads, shear forces, and bending moments in beams. It states that shear forces and bending moments are internal stress resultants that can be calculated from equations of equilibrium. Distributed loads cause shear forces to vary linearly or quadratically along the beam and bending moments to vary quadratically or cubically. Concentrated loads cause an abrupt change in shear force but no change in bending moment. Couples cause no change in shear force but an abrupt change in bending moment.
Structural Integrity Analysis: Chapter 3 Mechanical Properties of MaterialsIgor Kokcharov
Structural Integrity Analysis features a collection of selected topics on structural design, safety, reliability, redundancy, strength, material science, mechanical properties of materials, composite materials, welds, finite element analysis, stress concentration, failure mechanisms and criteria. The engineering approaches focus on understanding and concept visualization rather than theoretical reasoning. The structural engineering profession plays a key role in the assurance of safety of technical systems such as metallic structures, buildings, machines, and transport. The third chapter explains the engineering tests and fundamentals of mechanical properties of materials.
This document summarizes concepts related to torsion and the torsion of circular elastic bars. It discusses the assumptions made in analyzing torsion, including that shear strain varies linearly from the central axis. It also covers determining shear stress and torque using the polar moment of inertia for circular cross-sections. The relationships between applied torque, shear stress, shear strain, and angle of twist are defined. Stress concentrations and alternative differential equations approaches are also summarized.
This document discusses statically indeterminate structures and thermal stresses. It begins by defining statically indeterminate structures as those where the number of unknowns is greater than the number of equilibrium equations, requiring additional equations. It provides examples of compound bars made of two materials, where the deformations are equal and stresses can be calculated. It also discusses temperature stresses that develop when a material is prevented from expanding or contracting freely due to a temperature change. The temperature strain and stress formulas are provided. Several example problems are then solved to calculate stresses, deformations and loads for statically indeterminate structures and those subjected to temperature changes.
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.
This document describes different types of beams based on their end support, cross-section shape, equilibrium condition, and geometry. Beams can be simply supported, continuous, overhanging, cantilever, fixed, or trussed based on their end support. Their cross-section can be I-beams, T-beams, or C-beams. Based on equilibrium, beams are either statically determinate or indeterminate. A beam's geometry can be straight, curved, or tapered.
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 provides an overview of reinforced concrete design principles for civil engineers and construction managers. It discusses the aim of structural design according to BS 8110, describes the properties and composite action of reinforced concrete, explains limit state design methodology, and summarizes key elements like slabs, beams, columns, walls, and foundations. The document also covers material properties, stress-strain curves, failure modes, and general procedures for slab sizing and design.
Pure bending of curved bar (polar coordinate)Pratish Sardar
This document discusses pure bending of curved bars. It presents the assumptions made in analyzing curved members under bending, including that plane cross sections remain plane after bending. It defines key parameters like the radii of the inner and outer fibers. The stress function is introduced and the equations for radial and tangential stresses are given. The boundary conditions for a curved bar under pure bending are described. Standard relations are presented for determining the coefficients of the stress function based on the boundary conditions. Radial and circumferential stress equations are provided in terms of these coefficients. Finally, the document indicates it will provide numerical examples.
So far, all of the exercises presented in this module have been statically determinate, i.e. there have been enough equations of equilibrium available to solve for the unknowns. This final section will be concerned with statically indeterminate structures, and two methods used to solve these problems will be presented.
This document provides an introduction to strength of materials, including concepts of stress, strain, Hooke's law, stress-strain relationships, elastic constants, and factors of safety. It defines key terms like stress, strain, elastic limit, modulus of elasticity, and ductile and brittle material behavior. Examples of stress and strain calculations are provided for basic structural elements like rods, bars, and composite structures. The document also covers compound bars, principle of superposition, and effects of temperature changes.
This document discusses beam deflection. It begins by defining beam deflection and the factors that affect it, including bending moment, material properties, and shape properties. It then presents the general formula for calculating beam deflection using double integration of the bending moment equation. Examples are given of using boundary conditions to solve for deflection in simply supported beams, cantilever beams, and beams under various loading types. Common deflection formulas are also presented.
This document summarizes key concepts related to structural analysis including:
1) The effects of axial and eccentric loading on columns including direct stress, bending stress, and maximum/minimum stresses.
2) Maximum and minimum pressures at the base of dams and retaining walls including calculations of total water/earth pressure, eccentricity, and stability conditions.
3) Forces and stresses on chimneys and walls due to wind pressure including calculations of direct stress from self-weight, wind force, induced bending moment, and maximum/minimum stresses.
The document discusses bending stresses in beams. It describes how bending stresses are developed in beams to resist bending moments and shearing forces. The theory of pure bending is introduced, where only bending stresses are considered without the effect of shear. Equations for calculating bending stresses are derived based on the beam's moment of inertia, bending moment, and distance from the neutral axis. Several beam cross-section examples are provided to demonstrate how to calculate the maximum bending stress and section modulus.
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.
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 beam theory and provides equations for calculating the deflection and slope of beams under different loading conditions. It defines key terms like bending moment, radius of curvature, flexural stiffness, and provides equations relating these terms. Specifically, it gives the deflection and slope equations for a cantilever beam with a point load, cantilever with uniform load, simply supported beam with central point load, and simply supported beam with uniform load.
we select cantilever beam having I,C,T section and we select material cast iron, stainless steel, steel and analyze base upon modal and static analysis.we see here deformation,stress ,strain and based upon it we conclude.
This document provides information about beams used in structural engineering. It defines beams, discusses their structural characteristics like moment of inertia and stresses, and describes different types of beams including simply supported, fixed, cantilever, and trussed beams. It also covers beam design, applications in bridges and cranes, potential failure modes from plastic hinges, buckling or material failure, and methods to prevent failures like lateral restraints.
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.
This document provides an overview of column design and analysis. It defines columns and discusses their common uses in structures like buildings and bridges. Short columns fail through crushing, while long columns fail through buckling. Euler developed the first equation to analyze buckling in columns. The document discusses factors that influence a column's buckling capacity, like its effective length which depends on end support conditions. It presents design equations and factors for different column types (short, long, intermediate) and materials (steel). Safety factors are larger for columns than other members due to their importance for structural stability.
International Journal of Engineering and Science Invention (IJESI) inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
OUTLINE
introduction
classification
loads
materials used
Type of reinforcement
RCC
construction methods in RCC
Analysis and design
Detailing
Basic Rules
Site visit
video
This document discusses springs and columns. It defines springs as elastic bodies that can be compressed or stretched and return to their original shape. Springs store mechanical potential energy and come in many shapes. The document then discusses deflection of springs using strain energy and Castigliano's theorem. It also covers close-coiled helical springs, laminated springs, columns, struts, buckling, slenderness ratio, Euler's theory for buckling, and Euler's formula for critical load.
This document provides an overview of topics related to simple stresses and strains, including:
- Types of stresses and strains such as tensile, compressive, direct stress, and direct strain.
- Hooke's law and how stress is proportional to strain below the material's yield point.
- Stress-strain diagrams and key points such as the elastic region, yield point, and fracture point.
- Definitions of terms like working stress, factor of safety, Poisson's ratio, and elastic moduli.
- Examples of problems calculating stresses, strains, extensions, and deformations of simple structural members under various loads.
International Journal of Computational Engineering Research(IJCER) ijceronline
nternational Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
This document analyzes the effects of delamination length on stresses and natural frequencies of composite beams. It performs structural and vibration analyses on carbon fiber composite beams with varying delamination lengths using ANSYS software. The results show that displacements, stresses, and natural frequencies all increase as delamination length increases, indicating failure of the composite material at higher delamination lengths. Analyses are conducted using solid elements and shell elements with 5 layers to model the composite beams. In both cases, longer delamination lengths produce higher displacements and stresses. The conclusion is that composite materials fail when delamination increases, shown by increased displacements and stresses, and decreased natural frequencies.
The document describes an experiment to determine the Young's modulus of elasticity of mild steel, brass, and aluminum using a beam deflection method. The experiment involves applying various loads to beams made of each material and measuring the deflection with a dial gauge. Load-deflection graphs are plotted for each material, and the slope of the graphs is used to calculate the Young's modulus. The experiment found values of 1551 N/mm^2 for mild steel, 3675 N/mm^2 for brass, and 4963 N/mm^2 for aluminum. Precautions taken included carefully handling the dial gauge and ensuring accurate load placement and gauge positioning.
This document describes different types of beams based on their end support, cross-section shape, equilibrium condition, and geometry. Beams can be simply supported, continuous, overhanging, cantilever, fixed, or trussed based on their end support. Their cross-section can be I-beams, T-beams, or C-beams. Based on equilibrium, beams are either statically determinate or indeterminate. A beam's geometry can be straight, curved, or tapered.
Experimental study on strength and flexural behaviour of reinforced concrete ...IOSR Journals
Abstract: Strength and flexural behaviour of reinforced concrete beams using deflected structural steel
reinforcement and the conventional steel reinforcement are conducted in this study. The reinforcement quantity
of both categories was approximately equalised. Mild steel flats with minimum thickness and corresponding
width are deflected to possible extent in a parabolic shape and semi-circular shape are fabricated and used as
deflected structural steel reinforcement in one part, whereas the fabrication of ribbed tar steel circular bars as
conventional reinforcement on the another part of the experiment for comparison in the concrete beams. All the
beams had same dimensions and same proportions of designed mix concrete, were tested under two point
loading system. As the result of experiments, it is found that the inverted catenary flats and their ties, transfers
the load through arch action of steel from loading points towards the supports before reaching the bottom
fibre at the centre of the beam as intended earlier. Thereby the load carrying capacity and the ductility ratio
has being increased in deflected structural steel reinforced beams when compared with ribbed tar steel
reinforced concrete beams, it is also observed that the failure mode (collapse pattern)is safer.
Keywords --Arch profile, Conventional steel reinforcement, Cracks, Collapse, Deflected structural steel,
Ductility ratio.
Structural and vibration analysis of delaminated composite beamsijceronline
This document discusses structural and vibration analysis of composite beams with delaminations. It presents results from finite element analysis in ANSYS to analyze the effects of delamination length on stresses and natural frequencies of composite beams made from carbon fiber, Kevlar, and flouro polymer. The analyses were conducted on simply supported beams with single-edge delaminations of lengths 381mm and 400mm using solid elements and shell elements with 5 layers. The results show that increasing delamination length increases displacement, stress, and decreases natural frequencies, indicating composite materials are more likely to fail at higher delamination lengths.
This document discusses optimization techniques for shape, size, and topology optimization of a cantilever beam. It begins with an introduction to optimization and its applications in engineering design. It then discusses traditional optimization techniques like graphical methods, Johnson's method, and hill climbing. It also covers modern techniques like genetic algorithms, simulated annealing, and neural networks. The document provides detailed descriptions of how these various techniques can be applied to the design optimization of a cantilever beam.
Effect of Prestressing Force, Cable Profile and Eccentricity on Post Tensione...IRJET Journal
This document presents a finite element analysis of post-tensioned concrete beams using ANSYS software. It investigates the effect of prestressing force, cable profile, and eccentricity on the beam's response. Various cable profiles including straight, trapezoidal, parabolic, and sloping tendons were modeled at different eccentricities. The results from ANSYS were validated by comparing to analytical calculations. It was found that the cable profile, prestressing force, and eccentricity all influence the beam's deflections and stresses, and should be considered in design. The 3D finite element model in ANSYS was determined to be suitable for analyzing the effects of different design features on post-tensioned concrete beams.
The document discusses different types of structural elements used in building construction including beams, loads, supports, columns, and trusses. It describes several types of beams such as simply supported beams, continuous beams, overhanging beams, cantilever beams, and fixed beams. It also discusses different types of loads, supports, and how columns and trusses function structurally.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
Cantilever Beam
1. ABSTRACT
This study investigates the deflection and stress distribution in a long, slender cantilever beam of
uniform rectangular cross section made of linear elastic material properties that are homogeneous
and isotropic. The deflection of a cantilever beam is essentially a three dimensional problem. An
elastic stretching in one direction is accompanied by a compression in perpendicular directions.
In this project ,static and Modal analysis is a process to determine the stress ,strain and deformation.
vibration characteristics(naturalfrequencies and mode shapes)of a structure or a machine component while
it is being designed. It has become a major alternative to provide a helpful contribution in understanding
control of many vibration phenomena which encountered in practice.
In this work we compared the stress and natural frequency for different material having same I, C and T
cross-sectional beam.The cantilever beam is designed and analyzed in ANSYS.The cantilever beamwhich
is fixed at one end is vibrated to obtain the natural frequency, mode shapes and deflection with different
sections and materials.
2. I
LIST OF CONTENTS
Contents Page No.
CHAPTER 1 01-06
Introduction
1.1 Beam 01
1.2 Classification of Beams 02
1.3 Moment of Inertia 02
1.4 Stresses in Beams 02
1.5 General Shapes 04
1.6 Thin Walled Shapes 04
1.7 Cantilever Beam 05
1.8 Application 05
1.9 Advantages and Disadvantages 06
CHAPTER 2 08-09
Literature Review
2.1 Abstract 08
2.2 Introduction 08
2.3 Literature Review 08
2.4 Conclusion 09
CHAPTER 3 10-12
Introduction to CAD
3.1 Overview 10
3.2 Advantages 12
CHAPTER 4 13-15
Introduction CREO
4.1 Introduction 13
4.2 Advantages of CREO Software 14
4.3 CREO Parametric Modules 15
3. II
4.4 3D Models 15
CHAPTER 5 17-22
Introduction to FEM
5.1 Introduction 17
5.2 Introduction to ANSYS 18
5.3 Structural Analysis 19
5.4 Loads In a Structural Analysis 21
5.5 Modal Analysis 21
5.6 Uses of Modal Analysis 22
5.7 Random Vibration Analysis 22
CHAPTER 6 23-33
Structural Analysis of Cantilever Beam
6.1 Condition 1 I-Section 23
6.1.1 Material Steel 23
6.1.2 Material Stainless Steel 26
6.1.3 Material Iron 28
6.2 Condition 2 C-Section 29
6.2.1 Material Steel 29
6.3 Condition 3 T-Section 33
6.3.1 Material Steel 33
CHAPTER 7 34-37
Modal Analysis of Cantilever Beam
7.1 Condition 1 34
7.1.1 Material Steel 34
7.1.2 Material Stainless Steel 37
RESULT TABLES 39
CONCLUSION 41
REFERENCES 42
4. III
LIST OF FIGURES
Fig No Name Page No
1.1 Beam 01
4.1 Frames 3D Models 15-16
6.1 I-Section Models 23-24
6.4 I-Section Steel Material Structural Analysis 25-26
6.7 I-Section Stainless Steel Material Structural Analysis 27-28
6.10 I-Section Iron Material Structural Analysis 28-29
6.13 C-Section Models 29-30
6.16 C-Section Steel Material Structural Analysis 31-32
6.19 T-Section Steel Material Structural Analysis 33
7.1 I-Section Models 34-35
7.3 I-Section Steel Material Modal Analysis 36-37
7.5 I-Section Stainless Steel Material Modal Analysis 37-38
5. IV
LIST OF TABLES
Table No. Name of the Table Page No.
1 Static analysis results 39
2 Model analysis results 40
6. 1
CHAPTER 1
INTRODUCTION
1.1 BEAM
A beam is a structural element that is capable of withstanding load primarily by resisting
against bending. The bending force induced into the material of the beam as a result of the external
loads, own weight, span and external reactions to these loads is called a bending moment. Beams
are characterized by their profile (shape of cross-section), their length, and their material.
Beams are traditionally descriptions of building or civil engineering structural elements, but
smaller structures such as truck or automobile frames, machine frames, and other mechanical or
structural systems contain beam structures that are designed and analyzed in a similar fashion.
Figure 1.1 BEAM
A statically determinate beam, bending (sagging) under a uniformly distributed load
Overview
Historically beams were squared timbers but are also metal, stone, or combinations of wood and
metal such as a flitch beam. Beams generally carry vertical gravitational forces but can also be
used to carry horizontal loads (e.g., loads due to an earthquake or wind or in tension to resist rafter
thrust as a tie beam or (usually) compression as a collar beam). The loads carried by a beam are
transferred to columns, walls, or girders, which then transfer the force to adjacent
structural compression members. In light frame construction joists may rest on beams.
7. 2
In carpentry a beam is called a plate as in a sill plate or wall plate, beam as in a summer
beam or dragon beam.
1.2 CLASSIFICATION OF BEAMS BASED ON SUPPORTS
In engineering, beams are of several types:
1. Simply supported - a beam supported on the ends which are free to rotate and have no
moment resistance.
2. Fixed - a beam supported on both ends and restrained from rotation.
3. Over hanging - a simple beam extending beyond its support on one end.
4. Double overhanging - a simple beam with both ends extending beyond its supports on
both ends.
5. Continuous - a beam extending over more than two supports.
6. Cantilever - a projecting beam fixed only at one end.
7. Trussed - a beam strengthened by adding a cable or rod to form a truss.
1.3 Area moment of inertia
In the beam equation I is used to represent the second moment of area. It is commonly known as
the moment of inertia, and is the sum, about the neutral axis, of dA*r^2, where r is the distance
from the neutral axis, and dA is a small patch of area. Therefore, it encompasses not just how much
area the beam section has overall, but how far each bit of area is from the axis, squared. The greater
I is, the stiffer the beam in bending, for a given material.
1.4 Stress in beams
Internally, beams experience compressive, tensile and shear stresses as a result of the loads
applied to them. Typically, under gravity loads, the original length of the beam is slightly reduced
to enclose a smaller radius arc at the top of the beam, resulting in compression, while the same
original beam length at the bottom of the beam is slightly stretched to enclose a larger radius arc,
and so is under tension. The same original length of the middle of the beam, generally halfway
between the top and bottom, is the same as the radial arc of bending, and so it is under neither
8. 3
compression nor tension, and defines the neutral axis (dotted line in the beam figure). The beam,
or flexural member, is frequently encountered in structures and machines, and its elementary stress
analysis constitutes one of the more interesting facets of mechanics of materials. A beam is a
member subjected to loads applied transverse to the long dimension, causing the member to bend.
For example, a simply-supported beam loaded at its third-points will deform into the exaggerated
bent shape shown Before proceeding with a more detailed discussion of the stress analysis of
beams, it is useful to classify some of the various types of beams and loadings encountered in
practice. Beams are frequently classified on the basis of supports or reactions. A beam supported
by pins, rollers, or smooth surfaces at the ends is called a simple beam. A simple support will
develop a reaction normal to the beam, but will not produce a moment at the reaction. If either, or
both ends of a beam projects beyond the supports, it is called a simple beam with overhang. A
beam with more than simple supports is a continuous beam. A simple beam, a beam with overhang,
and a continuous beam. A cantilever beam is one in which one end is built into a wall or other
support so that the built-in end cannot move transversely or rotate. The built-in end is said to be
fixed if no rotation occurs and restrained if a limited amount of rotation occurs. The cantilever
beam, a beam fixed (or restrained) at the left end and simply supported near the other end (which
has an overhang) and a beam fixed (or restrained) at both ends, respectively. Cantilever beams and
simple beams have two reactions (two forces or one force and a couple) and these reactions can be
obtained from a free-body diagram of the beam by applying the equations of equilibrium. Such
beams are said to be statically determinate since the reactions can be obtained from the equations
of equilibrium. Continuous and other beams with only transverse loads, with more than two
reaction components are called statically indeterminate since there are not enough equations of
equilibrium to determine the reactions. Above the supports, the beam is exposed to shear stress.
There are some reinforced concrete beams in which the concrete is entirely in compression with
tensile forces taken by steel tendons. These beams are known as concrete beams, and are fabricated
to produce a compression more than the expected tension under loading conditions. High strength
steel tendons are stretched while the beam is cast over them. Then, when the concrete has cured,
the tendons are slowly released and the beam is immediately under eccentric axial loads. This
eccentric loading creates an internal moment, and, in turn, increases the moment carrying capacity
of the beam. They are commonly used on highway bridges.
9. 4
The primary tool for structural analysis of beams is the Euler–Bernoulli beam equation. Other
mathematical methods for determining the deflection of beams include "method of virtual work"
and the "slope deflection method". Engineers are interested in determining deflections because the
beam may be in direct contact with a brittle material such as glass. Beam deflections are also
minimized for aesthetic reasons. A visibly sagging beam, even if structurally safe, is unsightly and
to be avoided. A stiffer beam (high modulus of elasticity and high second moment of area)
produces less deflection.
Mathematical methods for determining the beam forces (internal forces of the beam and the forces
that are imposed on the beam support) include the "moment distribution method", the force
or flexibility method and the direct stiffness method.
1.5 Generalshapes
Most beams in reinforced concrete buildings have rectangular cross sections, but a more efficient
cross section for a beam is an I or H section which is typically seen in steel construction. Because
of the parallel axis theorem and the fact that most of the material is away from the neutral axis, the
second moment of area of the beam increases, which in turn increases the stiffness.
An I-beam is only the most efficient shape in one direction of bending: up and down looking at
the profile as an I. If the beam is bent side to side, it functions as an H where it is less efficient.
The most efficient shape for both directions in 2D is a box (a square shell) however the most
efficient shape for bending in any direction is a cylindrical shell or tube. But, for unidirectional
bending, the I or wide flange beam is superior.
Efficiency means that for the same cross sectional area (volume of beam per length) subjected to
the same loading conditions, the beam deflects less.
Other shapes, like L (angles), C (channels) or tubes, are also used in construction when there are
special require elements.
10. 5
1.6 Thin walled beams
A thin walled beam is a very useful type of beam (structure). The cross section of thin walled
beams is made up from thin panels connected among them to create closed or open cross sections
of a beam (structure). Typical closed sections include round, square, and rectangular tubes. Open
sections include I-beams, T-beams, L-beams, and so on. Thin walled beams exist because their
bending stiffness per unit cross sectional area is much higher than that for solid cross sections such
a rod or bar. In this way, stiff beams can be achieved with minimum weight. Thin walled beams
are particularly useful when the material is a composite laminates. Pioneer work on composite
laminates thin walled beams was done by Librescu.
1.7 CANTILEVER BEAM
A cantilever is a rigid structural element, such as a beam or a plate, anchored at only one end to a
(usually vertical) support from which it is protruding. Cantilevers can also be constructed with
trusses or slabs. When subjected to a structural load, the cantilever carries the load to the support
where it is forced against by a moment and shear stress.
Cantilever construction allows for overhanging structures without external bracing, in contrast to
constructions supported at both ends with loads applied between the supports, such as a simply
supported beam found in a post and lintel system.For a simple rectangular cantilever, the thickness,
length, and width of the beam determine the geometric shape. Each of these parameters affects
how a cantilever moves and bends. For example, a short cantilever is stiffer than a long cantilever
of the same material, width and thickness (see figure right). In macroapplications, short cantilevers
are used for balconies and longer cantilevers are used for diving boards. In microapplications, a
short cantilever (~ 10 microns) works best as a latch or a needle. A longer cantilever (~100
microns) works best as a transducer or sensor. However, there are applications where a long
cantilever (e.g. a 1.2 mm neural probe) requires the same rigidity as a 10 micron needle. In this
case, the other cantilever dimensions (width, thickness) and possibly the material would be
adjusted to provide the rigidity needed.
11. 6
1.8 APPLICATIONS
In bridges, towers, and buildings Cantilevers are widely found in construction, notably in cantilever
bridges and balconies (see corbel). In cantilever bridges the cantilevers are usually built as pairs, with
each cantilever used to support one end of a central section. The Forth Bridge in Scotland is an example
of a cantilever truss bridge. A cantilever in a traditionally timber framed building is called
a jetty or forebay. In the southern United States a historic barn type is the cantilever barn of log
construction.
Temporary cantilevers are often used in construction. The partially constructed structure creates a
cantilever, but the completed structure does not act as a cantilever. This is very helpful when
temporary supports, or falsework, cannot be used to support the structure while it is being built
(e.g., over a busy roadway or river, or in a deep valley). So some truss arch bridges (see Navajo
Bridge) are built from each side as cantilevers until the spans reach each other and are then jacked
apart to stress them in compression before final joining. Nearly all cable-stayed bridges are built
using cantilevers as this is one of their chief advantages. Many box girder bridges are
built segmentally, or in short pieces. This type of construction lends itself well to balanced
cantilever construction where the bridge is built in both directions from a single support.
These structures are highly based on torque and rotational equilibrium.
In an architectural application, Frank Lloyd Wright's Fallingwater used cantilevers to project large
balconies. The East Stand at Elland Road Stadium in Leeds was, when completed, the largest
cantilever stand in the world[2] holding 17,000 spectators. The roof built over the stands at Old
Trafford Football Ground uses a cantilever so that no supports will block views of the field. The
old, now demolished Miami Stadium had a similar roof over the spectator area. The largest
cantilever in Europe is located at St James' Park in Newcastle-Upon-Tyne, the home stadium
of Newcastle United F.C.
Less obvious examples of cantilevers are free-standing (vertical) radio towers without guy-wires,
and chimneys, which resist being blown over by the wind through cantilever action at their base.
12. 7
1.9 ADVANTAGES AND DISADVANTAGES
Advantages
Does not require a support on the opposite side (probably the main reason you would
ever have a cantilever beam).
Creates a negative bending moment, which can help to counteract a positive bending
moment created elsewhere. This is particular helpful in cantilevers with a backspan
where a uniform load on the backspan creates positive bending, but a uniform load on
the cantilever creates negative bending.
Disadvantages
Large deflections
Generally results in larger moments
You either need to have a fixed support, or have a backspan and check for uplift of the
far support.
13. 8
CHAPTER 2
LITERATURE REVIEW
A Review of Vibration of a cantileverBeam
Prof. Vyankatesh S. Kulkarni Department of Mechanical Engineering, Solapur University
/BIT/Barshi/India
2.1 Abstract :
Estimating damping in structure made of different materials (steel, brass, aluminum) and processes
still remains as one of the biggest challengers. All materials posses certain amount of internal
damping, which manifested as dissipation of energy from the system. This energy in a vibratory
system is either dissipated into heat or radiated away from the system. Material damping or internal
damping contributes to about 10-15% of total system damping. Cantilever beams of required size
& shape are prepared for experimental purpose & damping ratio is investigated. Damping ratio is
determined by half-power bandwidth method. It is observed that damping ratio is higher for steel
than brass than aluminum.
2.2 Introduction :
A wealth of literature exists in the area of vibrations of beams but while going through the literature
regarding material damping of cantilever beams it has been figured out that still a lot of work has
to be done regarding it. Usually whenever study of various materials has been done the focus of
researchers has been damping, mode shapes, resonant frequency, etc. but material damping has
not been paid much attention.
2.3 Literature Review:
1. H H Yoo and S H Shin [4] Vibration analysis of a rotating cantilever beam is an important and
peculiar subject of study in mechanical engineering. There are many engineering examples which
can be idealized as rotating cantilever beams such as turbine blades or turbo engine blades and
helicopter blades .For the proper design of the structures their vibration characteristics which are
natural frequencies and mode shapes should be well identified. Compared to the vibration
characteristics of non rotating structures those of rotating structures often vary significantly. The
variation results from the stretching induced by the centrifugal inertia force due to the rotational
14. 9
motion. The stretching causes the increment of the bending stiffness of the structure which
naturally results in the variation of natural frequencies and mode shapes. The equations of motion
of a rotating cantilever beam are derived based on a new dynamic modeling method. With the
coupling effect ignored the analysis results are consistent with the results obtained by the
conventional modelling method. A modal formulation method is also introduced in this study to
calculate the tuned angular speed of a rotating beam at which resonance occurs. 2. Mousa Rezaee
and Reza Hassannejad [14] derived a new analytical method for vibration analysis of a cracked
simply supported beam is investigated.
2.4 Conclusion:
The main objective of the present work is to study the vibration damping characteristics of three
materials i.e. steel, brass and aluminum. On the basis of present study following conclusions are
drawn: From the review it is evident that material damping is higher for steel in comparison with
brass and aluminum. The increase in material damping could be correlated to the stiffness of
materials. The damping ratio increases with decrease in thickness for each material. The natural
frequency decreases with decreases in thickness for each material. But it is viceversa in case of
length. The damping of specimen made up of aluminum was found to be lowest than either steel
or brass.
15. 10
CHAPTER 3
INTRODUCTION TO CAD
3.1 OVERVIEW:
Today’s industries cannot survive worldwide competition unless they introduce new products with
better quality (quality, Q), at lower cost (cost, C), and with shorter lead time (delivery, D).
Accordingly, they have tried to use the computer’s huge memory capacity, fast processing speed,
and user-friendly interactive graphics capabilities to automate and tie together otherwise
cumbersome and separate engineering or production tasks, thus reducing the time and cost of
product development and production. Computer-aided design (CAD), computer-aided
manufacturing (CAM), and computer-aided engineering (CAE) are the technologies used for this
purpose during the product cycle. Thus, to understand the role of CAD, CAM, and CAB, we need
to examine the various activities and functions that must be accomplished in the design and
manufacture of a product. These activities and functions are referred to as the product cycle. The
product cycle described by Zeid [1991] is presented here with minor modificationsThroughout the
history of our industrial society, many inventions have been patented and whole new technologies
have evolved. Perhaps the single development that has impacted manufacturing more quickly and
significantly than any previous technology is the digital computer.
Computers are being used increasingly for both design and detailing of engineering components
in the drawing office. Computer-aided design (CAD) is defined as the application of computers
and graphics software to aid or enhance the product design from conceptualization to
documentation. CAD is most commonly associated with the use of an interactive computer
graphics system, referred to as a CAD system. Computer-aided design systems are powerful tools
and in the mechanical design and geometric modeling of products and components.
Once the conceptual design has been developed, the analysis subprocess begins with analysis and
optimization of the design. An analysis model is derived first because the analysis subprocess is
16. 11
applied to the model rather than the design itself. Despite the rapid growth in the power and
availability of computers in engineering, the abstraction of analysis models will still be with us for
the foreseeable future. The analysis model is obtained by removing from the, design unnecessary
details, reducing dimensions, and recognizing and employing symmetry. Dimensional reduction,
for example, implies that a thin sheet of material is represented by an equivalent surface with a
thickness attribute or that a long slender region is represented by a line having cross-sectional
properties. Bodies with symmetries in their geometry and loading are usually analyzed by
considering a portion of the model. In fact, you have already practiced this abstraction process
naturally when you analyzed a structure in an elementary mechanics class. Recall that you always
start with sketching the structure in a simple shape before performing the actual analysis. Typical
of the analysis are stress analysis to verify the strength of the design, interference checking to
detect collision between components while they are moving in an assembly, and kinematic analysis
to check whether the machine to be used will provide the required motions. The quality of the
results obtained from these activities is directly related to and limited by the quality of the analysis
model chosen.
Designers generally use drawings to represent the object which they are designing, and to
communicate the design to others. Of course they will also use other forms of representation —
symbolic and mathematical models, and perhaps three-dimensional physical models — but the
drawing is arguably the most flexible and convenient of the forms of representation available.
Drawings are useful above all, obviously, for representing the geometrical form of the designed
object, and for representing its appearance. Hence the importance in computer-aided design (CAD)
of the production of visual images by computer, that is computer graphics. In the process of design,
technical drawings are used. Drawings explain the design and also establish the link between
design and manufacture. During the stage of design and detailing depend on the designers’ skill
and experience. Changes in previous designs take a long time because the drawings have to be
produced again. Computer-aided drawing is a technique where engineering drawings are produced
with the assistance of a computer and, as with manual drawing, is only the graphical means of
representing a design. Computer aided design, however, is a technique where the attributes of the
computer and those of the designer are blended together into a problem-solving team. When the
term CAD is used to mean computer-aided design it normally refers to a graphical system where
17. 12
components and assemblies can be modelled in three dimensions. The term design, however, also
covers those functions attributed to the areas of modelling and analysis. The acronym CADD is
more commonly used nowadays and stands for computer-aided draughting and design; a CADD
package is one which is able to provide all draughting facilities and some or all of those required
for the design process. Two-dimensional (2D) computer drawing is the representation of an object
in the single-view format which shows two of the three object dimensions or the mutiview format
where each view reveals two dimensions. In both cases, the database includes just two values for
each represented coordinate of the object. It can also be a pictorial representation if the database
contains X, Y coordinates only.
3.2 Advantages:
There are several good reasons for using a CAD system to support the engineering design function:
• To increase the productivity
• To improve the quality of the design
• To uniform design standards
• To create a manufacturing data base
• To eliminate inaccuracies caused by hand-copying of drawings and inconsistency between
• Drawings
18. 13
CHAPTER 4
INTRODUCTION TO CREO
4.1 Introduction:
PTC CREO, formerly known as Pro/ENGINEER, is 3D modeling software used in mechanical
engineering, design, manufacturing, and in CAD drafting service firms. It was one of the first 3D
CAD modeling applications that used a rule-based parametric system. Using parameters,
dimensions and features to capture the behavior of the product, it can optimize the development
product as well as the design itself.
The name was changed in 2010 from Pro/ENGINEER Wildfire to CREO. It was announced by
the company who developed it, Parametric Technology Company (PTC), during the launch of its
suite of design products that includes applications such as assembly modeling, 2D orthographic
views for technical drawing, finite element analysis and more.
PTC CREO says it can offer a more efficient design experience than other modeling software
because of its unique features including the integration of parametric and direct modeling in one
platform. The complete suite of applications spans the spectrum of product development, giving
designers options to use in each step of the process. The software also has a more user friendly
interface that provides a better experience for designers. It also has collaborative capacities that
make it easy to share designs and make changes.
There are countless benefits to using PTC CREO. We’ll take a look at them in this two-part series.
First up, the biggest advantage is increased productivity because of its efficient and flexible design
capabilities. It was designed to be easier to use and have features that allow for design processes
to move more quickly, making a designer’s productivity level increase.
Part of the reason productivity can be increased is because the package offers tools for all phases
of development, from the beginning stages to the hands-on creation and manufacturing. Late stage
changes are common in the design process, but PTC CREO can handle it. Changes can be made
that are reflected in other parts of the process.
19. 14
The collaborative capability of the software also makes it easier and faster to use. One of the
reasons it can process information more quickly is because of the interface between MCAD and
ECAD designs. Designs can be altered and highlighted between the electrical and mechanical
designers working on the project.
The time saved by using PTC CREO isn’t the only advantage. It has many ways of saving costs.
For instance, the cost of creating a new product can be lowered because the development process
is shortened due to the automation of the generation of associative manufacturing and service
deliverables.
PTC also offers comprehensive training on how to use the software. This can save businesses by
eliminating the need to hire new employees. Their training program is available online and in-
person, but materials are available to access anytime.
A unique feature is that the software is available in 10 languages. PTC knows they have people
from all over the world using their software, so they offer it in multiple languages so nearly anyone
who wants to use it is able to do so.
4.2 ADVANTAGES OF CREO PARAMETRIC SOFTWARE
Optimized for model-based enterprises
Increased engineer productivity
Better enabled concept design
Increased engineering capabilities
Increased manufacturing capabilities
Better simulation
Design capabilities for additive manufacturing
4.3 CREO parametric modules:
Sketcher
20. 15
Part modeling
Assembly
Drafting
4.4 3D models
I-section
I-Figure 4.1 3D MODEL OF I-SECTION
22. 17
CHAPTER 5
INTRODUCTION TO FEM
5.1 Introduction:
The Basic concept in FEA is that the body or structure may be divided into smaller elements of
finite dimensions called “Finite Elements”. The original body or the structure is then considered
as an assemblage of these elements connected at a finite number of joints called “Nodes” or “Nodal
Points”. Simple functions are chosen to approximate the displacements over each finite element.
Such assumed functions are called “shape functions”. This will represent the displacement with in
the element in terms of the displacement at the nodes of the element.
The Finite Element Method is a mathematical tool for solving ordinary and partial differential
equations. Because it is a numerical tool, it has the ability to solve the complex problems that can
be represented in differential equations form. The applications of FEM are limitless as regards the
solution of practical design problems.
Due to high cost of computing power of years gone by, FEA has a history of being used to
solve complex and cost critical problems. Classical methods alone usually cannot provide adequate
information to determine the safe working limits of a major civil engineering construction or an
automobile or an aircraft. In the recent years, FEA has been universally used to solve structural
engineering problems. The departments, which are heavily relied on this technology, are the
automotive and aerospace industry. Due to the need to meet the extreme demands for faster,
stronger, efficient and lightweight automobiles and aircraft, manufacturers have to rely on this
technique to stay competitive.
FEA has been used routinely in high volume production and manufacturing industries for many
years, as to get a product design wrong would be detrimental. For example, if a large manufacturer
had to recall one model alone due to a hand brake design fault, they would end up having to replace
up to few millions of hand brakes. This will cause a heavier loss to the company.
The finite element method is a very important tool for those involved in engineering design; it is
now used routinely to solve problems in the following areas.
Structural analysis
• Thermal analysis
• Vibrations and Dynamics
23. 18
• Buckling analysis
• Acoustics
• Fluid flow simulations
• Crash simulations
• Mold flow simulations
Nowadays, even the most simple of products rely on the finite element method for design
evaluation. This is because contemporary design problems usually can not be solved as accurately
& cheaply using any other method that is currently available. Physical testing was the norm in the
years gone by, but now it is simply too expensive and time consuming also.
5.2 INTRODUCTIONTO ANSYS
The ANSYS program is self contained general purpose finite element program developed
and maintained by Swason Analysis Systems Inc. The program contain many routines, all inter
related, and all for main purpose of achieving a solution to an an engineering problem by finite
element method.
ANSYS finite element analysis software enables engineers to perform the following tasks:
• Build computer models or transfer CAD models of structures, products, components, or
systems.
• Apply operating loads or other design performance conditions
• Study physical responses,such as stress levels, temperature distributions, or
electromagnetic fields
• Optimize a design early in the development process to reduce production costs.
• Do prototype testing in environments where it otherwise would be undesirable or
impossible
The ANSYS program has a compressive graphical user interface (GUI) that gives users easy,
interactive access to program functions, commands, documentation, and reference material. An
intuitive menu system helps users navigate through the ANSYS Program. Users can input data
using a mouse, a keyboard, or a combination of both. A graphical user interface is available
throughout the program, to guide new users through the learning process and provide more
24. 19
experienced users with multiple windows, pull-down menus, dialog boxes, tool bar and online
documentation.
5.3 STRUCTURALANALYSIS
Static analysis calculates the effects of steady loading conditions on a structure, while ignoring
inertia and damping effects, such as those caused by time-varying loads. A static analysis,
however, includes steady inertia loads (such as gravity and rotational velocity), and time-varying
loads that can be approximated as static equivalent loads (such as the static equivalent wind and
seismic loads commonly defined in many building codes). In order to conduct the analysis, both
the geometry of the structure and the actions and support conditions are idealised by means of an
adequate mathematical model, which must also roughly reflect the stiffness conditions of the cross-
sections, members, joints and interaction with the ground. The structural models must allow to
consider the effects of movements and deformations in those structures or part thereof, where
second-order effects increase the effects of the actions significantly. In certain cases, the model
must incorporate the following into its stiffness conditions:
- the non-linear response of the material outside the elastic analysis;
- the effects of shear lag in sections with wide flanges;
- the effects of local buckling in compressed sheet panels;
- the effects of the catenary (using a reduced modulus of elasticity, for example) and of
displacement on structures with cables;
- the shear deformability of certain structural members;
- the stiffness of the joints;
- interaction between the ground and the structure.
Where it is necessary to conduct dynamic analyses, the structural models must also consider the
properties of mass, stiffness, resistance and damping of each structural member, as well as the
mass of other, non-structural, members. Where it is appropriate to perform a quasi-static
approximation of the structure's dynamic effects in accordance with the codes or regulations in
force, such effects may be included in the static values of the actions, or dynamic amplification
factors equivalent to such static actions could even be applied. In some cases (e.g. vibrations
caused by wind or earthquake), the effects of the actions may be obtained from linear elastic
25. 20
analyses using the modal superposition method. Structural analyses for fire require specific models
that are considered. In some cases, the results of the structural analysis may undergo marked
variations regarding to possible fluctuations in some model parameters or in the design hypotheses
adopted. In such cases, the Designer shall perform a sensitivity analysis that allows to limit the
probable range of fluctuations in the structural response.
The content of this subsection only applies directly to linear members subjected to torsion where
the distance between points where there is no moment is equal to orgreater than 2.5 times its depth,
and the width is less than or equal to four times its depth, and the directrix is straight or curved.
The response of linear members to torsion, where the effects of distortion on the members may be
discounted, is the sum of two mechanisms: a) uniform or Saint-Venant torsion that only generates
shear stresses in the crosssection and the stiffness of which is characterised by the torsion modulus
It of the cross-section; b) non-uniform or warping torsion that generates both direct and shear stress
in the different sheet panels of the cross-section. Its stiffness remains characterised by its warping
modulus, Iw. The response of a member to torsion may be obtained through an elastic analysis that
incorporates the general equations for mixed torsion, depending on the static torsional magnitudes
of the cross-sections, It and Iw, the material deformation modulus, E and G, the connecting factors
for rotation and warping at the ends of the member, and the distribution of torsion action along it.
Alternatively, the structural analysis for torsion may be approached through finite elements models
for the part. It may be permitted for the effects of warping stress to be discounted in a suitably
approximate way, to analyse just the uniform torsion in members in the following cases: a)
members that have freedom to warp at their extremities and which are required solely for moment
at such extremities; b) members in which the warping module of the cross-section, Iw, is of
negligible or small magnitude in comparison with the torsion module, It. This is the case for the
following: – solid sections (round, square, rectangular, etc.); – open cross-sections made up of
rectangles that are sheared at a given point (angles, cross-shaped sections, single T units, etc.); –
closed cross-sections (tubes, single-cell or multi-cell boxes with no distortion, etc.). Additionally,
by way of simplification, it may be permitted for the effects of uniform torsion to be discounted,
and only analyse the warping stress, in the case of beams with thin-walled open sections such as
double T, U, H, Z sections, etc.
26. 21
5.4 LOADS IN A STRUCTURALANALYSIS
Static analysis is used to determine the displacements, stresses, strains, and forces in
structures or components caused by loads that do not induce significant inertia and damping
effects. Steady loading and response conditions are assumed; that is, the loads and the structure's
response are assumed to vary slowly with respect to time. The kinds of loading that can be applied
in a static analysis include:
• Externally applied forces and pressures
• Steady-state inertial forces (such as gravity or rotational velocity)
• Imposed (non-zero) displacements
• Temperatures (for thermal strain)
• Fluences (for nuclear swelling)
5.5 MODAL ANALYSIS
Any physical system can vibrate. The frequencies at which vibration naturally occurs, and the
modal shapes which the vibrating system assumes are properties of the system, and can be
determined analytically using Modal Analysis.
Modal analysis is the procedure of determining a structure's dynamic characteristics;
namely, resonant frequencies, damping values, and the associated pattern of structural deformation
called mode shapes. It also can be a starting point for another, more detailed, dynamic analysis,
such as a transient dynamic analysis, a harmonic response analysis, or a spectrum analysis.
Modal analysis in the ANSYS family of products is a linear analysis. Any nonlinearities,
such as plasticity and contact (gap) elements, are ignored even if they are defined. Modal analysis
can be done through several mode extraction methods: subspace, Block Lanczos, Power
Dynamics, Reduced, Unsymmetrical and Damped. The damped method allows you to include
damping in the structure. Theoretical modal analysis was developed during the 19th century. At
that stage, the analytical approaches were used to solve differential equations to determine the
modal parameters. During the last century, the theoretical modal analysis for the complex and large
systems made great progress with the fast development of the discretization technique (finite
element method) and computer techniques. Since then, numerical rather than analytical methods
have been commonly used in the modal analysis, called numerical modal analysis.
27. 22
5.6 USES OF MODAL ANALYSIS
Modal analysis is used to determine the natural frequencies and mode shapes of a structure. The
natural frequencies and mode shapes are important parameters in the design of a structure for
dynamic loading conditions. They are also required to do a spectrum analysis or a mode
superposition harmonic or transient analysis. Another useful feature is modal cyclic symmetry,
which allows reviewing the mode shapes of a cyclically symmetric structure by modeling just a
sector of it.
5.7 RANDOM VIBRATION ANALYSIS
A Random Vibration Analysis is a form of Spectrum Analysis.
•The spectrum is a graph of spectral value versus frequency that captures the intensity and
frequency content of time-history loads.
•Random vibration analysis is probabilistic in nature, because both input and output quantities
represent only the probability that they take on certain values
Random Vibration Analysis uses Power spectral density to quantify the loading.
• (PSD) is a statistical measure defined as the limiting mean-square value of a random variable. It
is used in random vibration analyses in which the instantaneous magnitudes of the response can
be specified only by probability distribution functions that show the probability of the magnitude
taking a particular value.
28. 23
CHAPTER 6
STRUCTURAL ANALYSIS OF CANTILEVER BEAM
6.1 CONDITION 1-I-SECTION
6.1.1 Material - STEEL
Save Creo Model as .iges format
→→Ansys → Workbench→ Select analysis system → static structural → double click
→→Select geometry → right click → import geometry → select browse →open part → ok
→→ select mesh on work bench → right click →edit
Figure 6.1 I-SECTION STEEL MATERIAL
Double click on geometry → select geometries → edit material
MATERIAL PROPERTIES OF STEEL
Density : 7850 kg/m3
Young’s modulus : 205000 Mpa
Poisson’s ratio : 0.3
29. 24
Select mesh on left side part tree → right click → generate mesh →
Figure 6.2 I-SECTION STEEL MATERIAL
Select static structural right click → insert → select pressure –0.096MPa
Figure 6.3 I-SECTION OF STEEL MATERIAL
Select displacement → select required area → click on apply →
Select solution right click → solve →
Solution- right click → insert → deformation → total
Solution right click → insert → strain → equivant (von-mises) →
Solution right click → insert → stress → equivant (von-mises) →
Right click on deformation → evaluate all result
30. 25
TOTAL DEFORMATION
Figure 6.4 TOTAL DEFORMATION OF STEEL MATERIAL
VON-MISES STRESS
Figure 6.5 VON-MISES STRESS OF STEEL MATERIAL
VON-MISES STRAIN
31. 26
Figure 6.6 VON-MISES STRAIN OF STEEL MATERIAL
6.1.2 MATERIAL- STAINLESS STEEL
MATERIAL PROPERTIES OF STEEL
Density : 5030 Kg/m3
Poisions ration : 294426 Mpa
Youngs modulus : 0.275
TOTAL DEFORMATION
Figure 6.7 TOTAL DEFORMATION OF STAINLESS STEEL
32. 27
VON-MISES STRESS
Figure 6.8 VON-MISES STRESS OF STAINLESS STEEL
VON-MISES STRAIN
Figure 6.9 VON-MISES STRAIN OF STAINLESS STEEL
33. 28
6.1.3 MATERIAL- Iron
6.4.1 MATERIAL PROPERTIES OF IRON
Density : 7874 Kg/m3
Young’s Modulus : 20400 Mpa
Poisons ratio : 0.29
TOTAL DEFORMATION
Figure 6.10 TOTAL DEFORMATION OF IRON MATERIAL
VON-MISES STRESS
Figure 6.11 VON-MISES STRESS IRON MATERIAL
34. 29
VON-MISES STRAIN
Figure 6.12 VON-MISES STRAIN OF IRON MATERIAL
6.2 CONDITION 2-C-SECTION
6.2.1 Material - STEEL
Imported model
Figure 6.13 MATERIAL STEEL
35. 30
Meshedmodel
Figure 6.14 MESHOD MODEL OF A STEEL MATERIAL
Force &displacement
Figure 6.15 FORCE & DISPLACEMENT OF A STEEL MATERIAL
36. 31
TOTAL DEFORMATION
Figure 6.16 TOTAL DEFORMATION OF A STEEL MATERIAL
VON-MISES STRESS
Figure 6.17 VON-MISES STRESS OF A STEEL MATERIAL
38. 33
6.3 CONDITION 1-T-SECTION
6.3.1 Material - STEEL
TOTAL DEFORMATION
Figure:6.19 TOTAL DEFORMATION OF STEEL MATERIAL
VON-MISES STRESS
Figure:6.20 VON-MISES STRESS OF STEEL MATERIAL
39. 34
CHAPTER 7
MODAL ANALYSIS OFCANTILEVER BEAM
7.1 CONDITION 1-I-SECTION
7.1.1 MATERIAL- STEEL
Save CREO Model as .IGES format
→→ANSYS → Workbench→ Select analysis system → model → double click
→→Select geometry → right click → import geometry → select browse →open part → OK
→→Select modal → right click →select edit → another window will be open
Figure 7.1 I- SECTION MODEL OF STEEL MATERIAL
MATERIAL PROPERTIES OF STEEL
Density : 7850kg/m3
Young’s modulus : 205000Mpa
Poisson’s ratio : 0.3
Select mesh on left side part tree → right click → generate mesh →
40. 35
Figure 7.2 I-SECTION MODEL OF STEEL MATERIAL
Select displacement → select required area → click on apply → Select solution right click →
solve →
Solution right click → insert → deformation → total deformation → mode 1
Solution right click → insert → deformation → total deformation2 → mode 2
Solution right click → insert → deformation → total deformation 3→ mode 3
Right click on deformation → evaluate all result
41. 36
TOTAL DEFORMATION 1:
Figure 7.3 TOTAL DEFORMATION OF STEEL MATERIAL
TOTAL DEFORMATION 2
Figure 7.4 TOTAL DEFORMATION 2 OF STEEL MATERIAL
42. 37
TOTAL DEFORMATION 3
Figure 7.5 TOTAL DEFORMATION 3 OF STEEL MATERIAL
7.1.2 MATERIAL– STAINLESS STEEL
TOTAL DEFORMATION 1
Figure 7.6 TOTAL DEFORMATION 1 OF STAINLESS STEEL
43. 38
TOTAL DEFORMATION 2
Figure 7.7 TOTAL DEFORMATION 2 OF STAINLESS STEEL
TOTAL DEFORMATION 3
Figure 7.8 TOTAL DEFORMATION 3 OF STAINLESS STEEL
45. 40
7.3 Modal analysis results
7.3.1 I-Section
Material Total Deformation
1(mm)
Total Deformation
2(mm)
Total Deformation
3(mm)
Steel 9.2158 11.7 9.06
Stainless steel 9.2752 11.8 9.12
Cast iron 9.6225 12.23 9.466
7.3.2 C-section
Material Total Deformation
1(mm)
Total Deformation
2(mm)
Total Deformation
3(mm)
Steel 9.9427 14.092 11.479
Stainless steel 10.012 14.192 11.553
Cast iron 10.37 14.691 11.987
7.3.3 T-Section
Material Total Deformation
1(mm)
Total Deformation
2(mm)
Total Deformation
3(mm)
Steel 13.151 13.465 10.829
Stainless steel 13.053 13.37 10.76
Cast iron 13.599 13.941 11.236
46. 41
CONCLUSION
In this work we compared the stress and natural frequency for different material having same I, C and T
cross-sectional beam.The cantilever beam is designed and analyzed in ANSYS.The cantilever beamwhich
is fixed at one end is vibrated to obtain the natural frequency, mode shapes and deflection with different
sections and materials.
By observing the static analysis the deformation and stress values are less for I-section cantilever
beam at cast iron material than steel and stainless steel.
By observing the modal analysis results the deformation and frequency values are less for I-section
cantilever beam more for T-section.
So it can be conclude the cast iron material is better material for cantilever beam in this type I-
section model.
47. 42
REFERENCES
[1] Chandradeep Kumar, Anjani Kumar Singh, Nitesh Kumar, Ajit Kumar, "Model Analysis and
Harmonic Analysis of Cantilever Beam by ANSYS" Global journal for research analysis, 2014,
Volume-3, Issue-9, PP:51- 55.
[2] Yuan, F.G. and R.E. Miller. A higher order finite element for laminated composite beams.
Computers & Structures, 14 (1990): 125-150.
[3] Dipak Kr. Maiti& P. K. Sinha. Bending and free vibration analysis of shear deformable
laminated composite beams by finite element method. Composite Structures, 29 (1994): 421- 431
[4]. DaDeppo, D. Introduction to Structural Mechanics and Analysis. Upper Saddle River, NJ:
Prentice-Hall, 1999.
[5]. Beer, F. P., and E. R. Johnston. Vector Mechanics for Engineers, Statics and Dynamics, 6th
ed. New York: McGraw-Hill, 1997.
[6[ Teboub Y, Hajela P. Free vibration of generally layered composite beams using symbolic
computations. Composite Structures, 33 (1995): 123– 34.
[7] Banerjee, J.R. Free vibration of axially loaded composite Timoshenko beams using the
dynamic stiffness matrix method. Computers & Structures, 69 (1998): 197-208 [6] Bassiouni AS,
Gad-Elrab RM, Elmahdy TH. Dynamic analysis for laminated composite beams. Composite
Structures, 44 (1999): 81–7.
[8]. Budynas, R. Advanced Strength and Applied Stress Analysis. 2d ed. New York: McGraw-
Hill, 1998.
[9] Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity. New York: Dover
Publications, 1944