A thorough review of the theoretical background of the invariant based approach to composite designs. The trace material property was explained, along with several topics from linear elasticity theory. The MATLAB implementation of a tool capable of determining the laminate stiffness properties from the lamina properties and stacking sequence was also presented. For access to the tool please contact the author: filipegiesteira@outlook.com
Assignment developed in the scope of the Composite Systems course, lecture at FEUP (Faculdade de Engenharia da Universidade do Porto).
This document discusses photoelasticity, which is a stress analysis technique that uses the relative retardation between two components of light passing through a photoelastic model. Key points:
1) Photoelasticity measures the difference in refractive indices along principal stress directions in a stressed transparent model, allowing calculation of principal stresses and strains.
2) When unstressed, photoelastic models are isotropic, but stress causes temporary birefringence by changing refractive indices along stress directions.
3) Relative retardation between light components is measured using polariscopes and relates to stress difference through the stress optic law and material properties.
4) Isoclinics and isochromatics appear as loci of points with specific principal stress orientations
This document provides an introduction to the theory of plates, which are structural elements that are thin and flat. It defines what is meant by a thin plate and discusses different plate classifications based on thickness. The document derives the basic equations that describe plate behavior by taking advantage of the plate's thin, planar character. It also discusses three-dimensional considerations like stress components, equilibrium, strain and displacement for putting the plate theory into context.
The document discusses isoparametric finite elements. It defines isoparametric, superparametric, and subparametric elements. It provides examples of shape functions for 4-noded rectangular, 6-noded triangular, and 8-noded rectangular isoparametric elements. It also discusses coordinate transformation from the natural to global coordinate system using these shape functions and calculating the Jacobian.
This document provides an overview of laminated composite materials and refined plate theories used to model their behavior. It discusses how classical plate theory (CPT) and first-order shear deformation theory (FSDT) have limitations for thick laminated composites due to neglecting transverse shear effects. Higher-order theories like trigonometric shear deformation theory (TSDT), hyperbolic shear deformation theory (HSDT), and second-order shear deformation theory (SSDT) are introduced to address these limitations. The objectives of the study are to develop new refined theories, establish their credibility by applying them to static flexure problems, and obtain results for laminated beams and plates under various loadings not widely available in literature.
Maximum principal stress theory.
Maximum shear stress theory.
Maximum shear strain theory.
Maximum strain energy theory.
Maximum shear strain energy theory.
To determine the ductility of mild steel specimens using a three-point bend test. The test involves placing steel bar specimens on a bend test machine with supports 8 times the bar diameter apart and a handle 5 times the bar diameter above the supports. A load is applied until the bar bends at 180 degrees, though some spring back was observed. No cracks were observed, indicating the material is suitable for use. The three-point bend test provides a simple way to evaluate materials' ability to resist cracking during bending.
The document discusses fatigue failure and fatigue analysis. It begins by explaining that fatigue failure starts with a crack, usually at a stress concentration, which then propagates until sudden fracture. It then provides examples of fatigue failures and discusses different fatigue analysis methods. The key points are:
- Fatigue failure results from repeated or fluctuating stresses that are lower than the material's ultimate strength.
- It can be analyzed using stress-life, strain-life, or fracture mechanics methods, with stress-life most common for high-cycle fatigue.
- The stress-life approach estimates fatigue strength (Sf) based on stress levels and uses modifying factors to account for real-world differences from test specimens.
This document discusses photoelasticity, which is a stress analysis technique that uses the relative retardation between two components of light passing through a photoelastic model. Key points:
1) Photoelasticity measures the difference in refractive indices along principal stress directions in a stressed transparent model, allowing calculation of principal stresses and strains.
2) When unstressed, photoelastic models are isotropic, but stress causes temporary birefringence by changing refractive indices along stress directions.
3) Relative retardation between light components is measured using polariscopes and relates to stress difference through the stress optic law and material properties.
4) Isoclinics and isochromatics appear as loci of points with specific principal stress orientations
This document provides an introduction to the theory of plates, which are structural elements that are thin and flat. It defines what is meant by a thin plate and discusses different plate classifications based on thickness. The document derives the basic equations that describe plate behavior by taking advantage of the plate's thin, planar character. It also discusses three-dimensional considerations like stress components, equilibrium, strain and displacement for putting the plate theory into context.
The document discusses isoparametric finite elements. It defines isoparametric, superparametric, and subparametric elements. It provides examples of shape functions for 4-noded rectangular, 6-noded triangular, and 8-noded rectangular isoparametric elements. It also discusses coordinate transformation from the natural to global coordinate system using these shape functions and calculating the Jacobian.
This document provides an overview of laminated composite materials and refined plate theories used to model their behavior. It discusses how classical plate theory (CPT) and first-order shear deformation theory (FSDT) have limitations for thick laminated composites due to neglecting transverse shear effects. Higher-order theories like trigonometric shear deformation theory (TSDT), hyperbolic shear deformation theory (HSDT), and second-order shear deformation theory (SSDT) are introduced to address these limitations. The objectives of the study are to develop new refined theories, establish their credibility by applying them to static flexure problems, and obtain results for laminated beams and plates under various loadings not widely available in literature.
Maximum principal stress theory.
Maximum shear stress theory.
Maximum shear strain theory.
Maximum strain energy theory.
Maximum shear strain energy theory.
To determine the ductility of mild steel specimens using a three-point bend test. The test involves placing steel bar specimens on a bend test machine with supports 8 times the bar diameter apart and a handle 5 times the bar diameter above the supports. A load is applied until the bar bends at 180 degrees, though some spring back was observed. No cracks were observed, indicating the material is suitable for use. The three-point bend test provides a simple way to evaluate materials' ability to resist cracking during bending.
The document discusses fatigue failure and fatigue analysis. It begins by explaining that fatigue failure starts with a crack, usually at a stress concentration, which then propagates until sudden fracture. It then provides examples of fatigue failures and discusses different fatigue analysis methods. The key points are:
- Fatigue failure results from repeated or fluctuating stresses that are lower than the material's ultimate strength.
- It can be analyzed using stress-life, strain-life, or fracture mechanics methods, with stress-life most common for high-cycle fatigue.
- The stress-life approach estimates fatigue strength (Sf) based on stress levels and uses modifying factors to account for real-world differences from test specimens.
1) The document discusses stresses in thin and thick cylinders, including circumferential (hoop), longitudinal, and radial stresses. It also covers principal stresses.
2) Formulas are provided for calculating wall thickness based on tangential stress in thin cylinders, and Lame's equation is introduced for thick cylinders.
3) Additional concepts covered include stresses in spherical vessels, pre-stressing techniques like autofrettage to increase pressure capacity, and stresses in cylinders under external pressure or combined internal and shrink pressures.
Fatigue is a type of failure caused by fluctuating stresses that are lower than the material's yield stress. It results from the initiation and propagation of cracks under cyclic stresses. Fatigue failure can occur suddenly in ductile materials without prior plastic deformation. The S-N curve relates the cyclic stress amplitude to the number of cycles until failure. It may show a fatigue limit below which failure will not occur even after an infinite number of cycles for some materials like steels. Non-ferrous materials typically do not have a fatigue limit.
This document discusses two-dimensional vector variable problems in structural mechanics. It describes plane stress, plane strain and axisymmetric problems, and provides the stress-strain relations for materials under these conditions. It also discusses thin structures like disks and long prismatic shafts. Additionally, it covers dynamic analysis and vibration of structures, describing free vibration, forced vibration and types of vibration. Equations of motion are developed using Lagrange's approach and the weak form method. Mass and stiffness matrices for axial rod and beam elements are also presented.
Strength of Materials Lecture - 2
Elastic stress and strain of materials (stress-strain diagram)
Mehran University of Engineering and Technology.
Department of Mechanical Engineering.
This document contains formulas and equations related to finite element analysis (FEA) for one-dimensional structural and heat transfer problems. It includes formulas for weighted residual methods, Ritz method, beam deflection and stress, springs, one-dimensional bars and frames, and one-dimensional heat transfer through walls and fins. Displacement functions, stiffness matrices, thermal loads, and conduction/convection equations are provided for linear and quadratic elements undergoing static structural and thermal analysis.
This document provides information about the Solid Mechanics course ME 302 taught by Dr. Nirmal Baran Hui at NIT Durgapur in West Bengal, India. It lists four required textbooks for the course and provides a detailed syllabus covering topics like stress, strain, elasticity, bending, deflection, columns, torsion, pressure vessels, combined loadings, springs, and failure theories. The document also includes examples of lecture content on stress analysis, stresses on oblique planes, and material subjected to pure shear.
Presentation for Fiber Composites course. Outlines the failure theories used in composite failure analysis and methods to design composite materials based on these failure theories.
Fatigue occurs when a material is subjected to repeated loading and unloading. It causes failure from crack initiation and propagation even when stresses are below the yield strength of the material. Fatigue was first observed in railroad and bridge components that cracked under repeated loading. Fatigue failure can occur suddenly and without warning in metals, plastics, rubbers, and concrete used in applications with rotating or fluctuating stresses like aircraft wings, springs, and pipes conveying fluid. The number of cycles to failure depends on the stress range and mean stress based on stress-life (S-N) curves, which can be corrected using the Goodman diagram for different stress ratios. Crack propagation rates under cyclic loading can be modeled in three regions based on stress
This document provides an introduction to fracture mechanics. It discusses different types of brittle and ductile fracture, modes of failure, energy release rate and crack resistance, crack growth, stress intensity factor, and the J-integral. It also mentions a case study on liberty ship failures and provides references for further reading. The key topics covered are the assumptions of fracture mechanics, using energy-based approaches like compliance and strain energy to analyze crack growth, and stress intensity factors which characterize how potent a crack is under different loading conditions.
Some basic defintions of the topics used in Strength of Materials subject. Pictorial presentation is more than details. Many examples are provided as well.
The document discusses brittle coating methods for experimental stress analysis. Brittle coatings crack under tensile stresses, allowing stresses in specimens to be determined. Key points:
- A thin brittle coating is applied to the specimen surface. As the specimen is loaded, strains are transmitted to the coating, causing cracks perpendicular to maximum tensile stresses.
- Coating crack patterns indicate stress directions. Multiple crack patterns can form under different stress conditions.
- Refrigeration and load relaxation techniques can induce cracks in low-stress regions.
- Fine coating cracks are detected using oblique light or Statiflux particle methods.
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
The document discusses different types of strain gauges used to measure strain. It describes how strain gauges work by changing electrical resistance proportionally to strain. Common types include metallic wire or foil grids bonded to a backing material. Materials like constantan/advance are often used due to properties like self-temperature compensation and linear strain sensitivity. Unbounded wire strain gauges consist of tensioned wires connected to a Wheatstone bridge circuit to measure strain through changes in electrical resistance. Strain gauges are widely applied to experimental stress analysis of structures, machines, vehicles and more.
This document discusses stress-strain diagrams and mechanical properties of materials. It includes 8 figures showing stress-strain diagrams for different material types (ductile, elastic, plastic, brittle) as well as diagrams for resilience, toughness, and creep. The document defines key points on stress-strain diagrams like proportional limit, elastic limit, yield limit, ultimate point, and fracture point. It also describes material properties including elasticity, plasticity, stiffness, ductility, brittleness, malleability, resilience, toughness, and creep. The document serves to explain stress-strain behavior and mechanical characteristics.
Saint-Venant's principle states that the stresses and strains far away from the load application point are unaffected by the exact nature of the load or its application method, but only depend on the resultant load magnitude and application area. Stress concentrations occur where the cross-sectional area changes abruptly, like holes, notches, or threads, and cause local stress values much higher than the average stress. The stress concentration factor K is used to relate the maximum stress σmax to the average stress σave in a cross-section. Design engineers use stress concentration factors and allowable stress values to determine if a given load will exceed the material's strength at stress concentration locations.
1. Cylinders are commonly used in engineering to transport or store fluids and are subjected to internal fluid pressures. This induces three stresses on the cylinder wall - circumferential, longitudinal, and radial.
2. For thin cylinders where the wall thickness is less than 1/20 the diameter, the radial stress can be neglected. Equations are derived to calculate the circumferential and longitudinal stresses based on the internal pressure, diameter, and wall thickness.
3. Sample problems are worked out applying the equations to example thin-walled cylinders under internal pressure, finding stresses, strains, and changes in dimensions.
This document discusses the 3-point flexural test, which measures the flexural properties of materials. In a 3-point flexural test, a specimen is placed on two supporting pins and a loading pin is applied in the middle. Calculations are performed to determine flexural stress, strain, and modulus based on the load and deflection measurements. The test provides values for modulus of elasticity in bending, flexural stress, flexural strain, and flexural stress-strain response. It is a common test for evaluating a material's stiffness when flexed.
1. The document discusses four common failure theories used in engineering: maximum shear stress theory, maximum principal stress theory, maximum normal strain theory, and maximum shear strain theory.
2. It provides details on the maximum shear stress (Tresca) theory, which states that failure occurs when the maximum shear stress equals the yield point stress under simple tension.
3. An example problem is presented involving determining the required diameter of a circular shaft using the maximum shear stress theory with a safety factor of 3, given the material properties and loads on the shaft.
The summary is:
1. The singular value decomposition (SVD) provides a factorization of any matrix A as A = UΣVT, where U and V are orthogonal matrices and Σ is a diagonal matrix of singular values.
2. The SVD exists for any matrix A by considering the eigendecompositions of the positive semidefinite matrices ATA and AAT.
3. The SVD is useful for finding least squares solutions to overdetermined or underdetermined systems of linear equations, by using the pseudoinverse of A.
Avionics 738 Adaptive Filtering at Air University PAC Campus by Dr. Bilal A. Siddiqui in Spring 2018. This lecture covers background material for the course.
1) The document discusses stresses in thin and thick cylinders, including circumferential (hoop), longitudinal, and radial stresses. It also covers principal stresses.
2) Formulas are provided for calculating wall thickness based on tangential stress in thin cylinders, and Lame's equation is introduced for thick cylinders.
3) Additional concepts covered include stresses in spherical vessels, pre-stressing techniques like autofrettage to increase pressure capacity, and stresses in cylinders under external pressure or combined internal and shrink pressures.
Fatigue is a type of failure caused by fluctuating stresses that are lower than the material's yield stress. It results from the initiation and propagation of cracks under cyclic stresses. Fatigue failure can occur suddenly in ductile materials without prior plastic deformation. The S-N curve relates the cyclic stress amplitude to the number of cycles until failure. It may show a fatigue limit below which failure will not occur even after an infinite number of cycles for some materials like steels. Non-ferrous materials typically do not have a fatigue limit.
This document discusses two-dimensional vector variable problems in structural mechanics. It describes plane stress, plane strain and axisymmetric problems, and provides the stress-strain relations for materials under these conditions. It also discusses thin structures like disks and long prismatic shafts. Additionally, it covers dynamic analysis and vibration of structures, describing free vibration, forced vibration and types of vibration. Equations of motion are developed using Lagrange's approach and the weak form method. Mass and stiffness matrices for axial rod and beam elements are also presented.
Strength of Materials Lecture - 2
Elastic stress and strain of materials (stress-strain diagram)
Mehran University of Engineering and Technology.
Department of Mechanical Engineering.
This document contains formulas and equations related to finite element analysis (FEA) for one-dimensional structural and heat transfer problems. It includes formulas for weighted residual methods, Ritz method, beam deflection and stress, springs, one-dimensional bars and frames, and one-dimensional heat transfer through walls and fins. Displacement functions, stiffness matrices, thermal loads, and conduction/convection equations are provided for linear and quadratic elements undergoing static structural and thermal analysis.
This document provides information about the Solid Mechanics course ME 302 taught by Dr. Nirmal Baran Hui at NIT Durgapur in West Bengal, India. It lists four required textbooks for the course and provides a detailed syllabus covering topics like stress, strain, elasticity, bending, deflection, columns, torsion, pressure vessels, combined loadings, springs, and failure theories. The document also includes examples of lecture content on stress analysis, stresses on oblique planes, and material subjected to pure shear.
Presentation for Fiber Composites course. Outlines the failure theories used in composite failure analysis and methods to design composite materials based on these failure theories.
Fatigue occurs when a material is subjected to repeated loading and unloading. It causes failure from crack initiation and propagation even when stresses are below the yield strength of the material. Fatigue was first observed in railroad and bridge components that cracked under repeated loading. Fatigue failure can occur suddenly and without warning in metals, plastics, rubbers, and concrete used in applications with rotating or fluctuating stresses like aircraft wings, springs, and pipes conveying fluid. The number of cycles to failure depends on the stress range and mean stress based on stress-life (S-N) curves, which can be corrected using the Goodman diagram for different stress ratios. Crack propagation rates under cyclic loading can be modeled in three regions based on stress
This document provides an introduction to fracture mechanics. It discusses different types of brittle and ductile fracture, modes of failure, energy release rate and crack resistance, crack growth, stress intensity factor, and the J-integral. It also mentions a case study on liberty ship failures and provides references for further reading. The key topics covered are the assumptions of fracture mechanics, using energy-based approaches like compliance and strain energy to analyze crack growth, and stress intensity factors which characterize how potent a crack is under different loading conditions.
Some basic defintions of the topics used in Strength of Materials subject. Pictorial presentation is more than details. Many examples are provided as well.
The document discusses brittle coating methods for experimental stress analysis. Brittle coatings crack under tensile stresses, allowing stresses in specimens to be determined. Key points:
- A thin brittle coating is applied to the specimen surface. As the specimen is loaded, strains are transmitted to the coating, causing cracks perpendicular to maximum tensile stresses.
- Coating crack patterns indicate stress directions. Multiple crack patterns can form under different stress conditions.
- Refrigeration and load relaxation techniques can induce cracks in low-stress regions.
- Fine coating cracks are detected using oblique light or Statiflux particle methods.
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
The document discusses different types of strain gauges used to measure strain. It describes how strain gauges work by changing electrical resistance proportionally to strain. Common types include metallic wire or foil grids bonded to a backing material. Materials like constantan/advance are often used due to properties like self-temperature compensation and linear strain sensitivity. Unbounded wire strain gauges consist of tensioned wires connected to a Wheatstone bridge circuit to measure strain through changes in electrical resistance. Strain gauges are widely applied to experimental stress analysis of structures, machines, vehicles and more.
This document discusses stress-strain diagrams and mechanical properties of materials. It includes 8 figures showing stress-strain diagrams for different material types (ductile, elastic, plastic, brittle) as well as diagrams for resilience, toughness, and creep. The document defines key points on stress-strain diagrams like proportional limit, elastic limit, yield limit, ultimate point, and fracture point. It also describes material properties including elasticity, plasticity, stiffness, ductility, brittleness, malleability, resilience, toughness, and creep. The document serves to explain stress-strain behavior and mechanical characteristics.
Saint-Venant's principle states that the stresses and strains far away from the load application point are unaffected by the exact nature of the load or its application method, but only depend on the resultant load magnitude and application area. Stress concentrations occur where the cross-sectional area changes abruptly, like holes, notches, or threads, and cause local stress values much higher than the average stress. The stress concentration factor K is used to relate the maximum stress σmax to the average stress σave in a cross-section. Design engineers use stress concentration factors and allowable stress values to determine if a given load will exceed the material's strength at stress concentration locations.
1. Cylinders are commonly used in engineering to transport or store fluids and are subjected to internal fluid pressures. This induces three stresses on the cylinder wall - circumferential, longitudinal, and radial.
2. For thin cylinders where the wall thickness is less than 1/20 the diameter, the radial stress can be neglected. Equations are derived to calculate the circumferential and longitudinal stresses based on the internal pressure, diameter, and wall thickness.
3. Sample problems are worked out applying the equations to example thin-walled cylinders under internal pressure, finding stresses, strains, and changes in dimensions.
This document discusses the 3-point flexural test, which measures the flexural properties of materials. In a 3-point flexural test, a specimen is placed on two supporting pins and a loading pin is applied in the middle. Calculations are performed to determine flexural stress, strain, and modulus based on the load and deflection measurements. The test provides values for modulus of elasticity in bending, flexural stress, flexural strain, and flexural stress-strain response. It is a common test for evaluating a material's stiffness when flexed.
1. The document discusses four common failure theories used in engineering: maximum shear stress theory, maximum principal stress theory, maximum normal strain theory, and maximum shear strain theory.
2. It provides details on the maximum shear stress (Tresca) theory, which states that failure occurs when the maximum shear stress equals the yield point stress under simple tension.
3. An example problem is presented involving determining the required diameter of a circular shaft using the maximum shear stress theory with a safety factor of 3, given the material properties and loads on the shaft.
The summary is:
1. The singular value decomposition (SVD) provides a factorization of any matrix A as A = UΣVT, where U and V are orthogonal matrices and Σ is a diagonal matrix of singular values.
2. The SVD exists for any matrix A by considering the eigendecompositions of the positive semidefinite matrices ATA and AAT.
3. The SVD is useful for finding least squares solutions to overdetermined or underdetermined systems of linear equations, by using the pseudoinverse of A.
Avionics 738 Adaptive Filtering at Air University PAC Campus by Dr. Bilal A. Siddiqui in Spring 2018. This lecture covers background material for the course.
1. The document discusses barriers to scaling electronic structure methods to large systems, such as the inability of sparse matrix multiplication kernels to access strong parallel scaling and entrenched data structures that limit innovation.
2. It proposes a fast, generic, and data local N-body solver approach using new mathematics that is not constrained by row-column data structures and allows a single programming model.
3. Key aspects of this approach include exploiting locality in higher dimensional product volumes through techniques like occlusion-culling, resolving identity iteratively to compress matrices by orders of magnitude, and developing optimized sparse matrix multiplication kernels.
This document summarizes key concepts from a lecture on linear algebra:
1) It defines terms like linear combinations, linear independence, orthonormal vectors, eigenvalues, and eigendecomposition as they relate to vectors and matrices.
2) It describes how to solve least squares problems by exploiting properties of positive semidefinite matrices like their eigendecomposition.
3) Solving a least squares problem for a positive semidefinite matrix M can be reduced to solving a simpler problem involving its eigenvectors and eigenvalues.
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
A Survey of Statistical Properties of Random Regular Graphs to Study Existing...Amin Shahnazari
This document summarizes research on phase transitions in random regular graphs. It discusses several types of phase transitions, including metal-insulator transitions, wave function localization, and Anderson localization. Random matrix theory is also covered as a way to characterize the extended and localized phases. The document examines localization in both real and Fock space, and how properties like the density of states and level spacing statistics can indicate phase transitions. It was written by Amin Shahnazari Zazerani from the University of Isfahan under the guidance of Dr. Mohsen Amini.
This document provides an overview of simulation modeling. It defines a system as any set of interrelated components acting together to achieve a common objective. A model represents the structure of a real system through simplification, abstraction, and assumptions. Simulation is the process of running a computer model of a real system to study or experiment with it. There are different types of simulations depending on whether changes are continuous or discrete over time and whether aspects are deterministic or stochastic. Monte Carlo simulation uses random sampling to approximate expectations while discrete event simulation models systems as sequences of discrete events over time. Examples provided include using Monte Carlo to estimate pi and modeling a single machine system in discrete event simulation software.
A Stabilized Finite Element Dynamic Overset Method for the Navier-Stokes Equa...Chao Liu
This document summarizes research on developing an overset finite element method for solving the Navier-Stokes equations. The key questions addressed are whether overset grids would deteriorate solution accuracy for stabilized finite element methods, how breaking a domain into overlapping sub-domains impacts solver performance, and how to implement dynamic grid assembly and parallel solving. Manufactured solutions and test cases involving oscillating wings and turbulent flow over a fighter jet demonstrate the overset method maintains accuracy compared to single grids and enables breaking domains across components for improved performance.
The document describes a self-study course on finite element methods for structural analysis developed by Dr. Naveen Rastogi. The course is intended as a 3-credit, semester-long course for senior undergraduate and graduate engineering students. It covers foundational concepts of finite element analysis including shape functions, element stiffness matrices, and static/dynamic/thermal analysis of structures. Practice problems are provided to help students gain a better understanding of the subject matter.
The document discusses developing quantitative structure-activity relationship (QSAR) models to predict the biological responses of nanomaterials. It describes using descriptors of pristine and weathered nanomaterials, as well as experimental parameters, to develop linear regression models between descriptors and responses. Partial least squares regression is used to handle correlations between descriptors. The data is also analyzed using k-means clustering to identify separate descriptor clusters, and QSAR models are developed for each cluster to improve predictions. The resulting models could then be used to predict responses of emerging nanomaterials based on their similarity to existing clusters.
Tensor Eigenvectors and Stochastic ProcessesAustin Benson
This document provides an overview of a tutorial on tensor eigenvectors and stochastic processes. The tutorial consists of 6 acts that cover basic tensor notation and operations, motivating applications involving tensors, a review of stochastic processes and Markov chains, introducing spacey random walks as stochastic processes, the theory of spacey random walks, and applications of spacey random walks. The goal is to interpret tensor objects from a stochastic perspective using concepts like random walks on graphs and higher-order Markov chains.
Probabilistic Models of Time Series and SequencesZitao Liu
Tutorial on Probabilistic Models of Time Series and Sequences. Hidden Markov Models. Linear Dynamical Systems. Forward/backward algorithm. Kalman Filtering. Kalman Smoothing. Viterbi algorithm. Baum-Welch algorithm. Learning HMM. Learning LDS.
The document discusses plate bending theory and the stress-strain hypothesis. Some key points:
1) Plate bending theory studies the behavior of thin, flat plates under external loads and is based on the assumption that plates are very thin compared to their other dimensions and undergo small deformations.
2) The stress-strain hypothesis relates the stress in a plate to the strain it undergoes, assuming stress is proportional to strain via the modulus of elasticity. It also assumes normal stress is proportional to curvature and shear stress is proportional to the rate of change of curvature.
3) The stress-strain hypothesis allows derivation of equations relating external loads on a plate to resulting deformations and stresses, using the principle of virtual work
Analysis of large scale spiking networks dynamics with spatio-temporal constr...Hassan Nasser
Recent experimental advances have made it possible to record up to several hundreds of neurons simultaneously in the cortex or in the retina. Analysing such data requires mathematical and numerical methods to describe the spatio-temporal correlations in population activity. This can be done thanks to Maximum Entropy method. Here, a crucial parameter is the product NxR where N is the number of neurons and R the memory depth of correlations (how far in the past does the spike activity affects the current state). Standard statistical mechanics methods are limited to spatial correlation structure with
R = 1 (e.g. Ising model) whereas methods based on transfer matrices, allowing the analysis of spatio-temporal correlations, are limited to NR = 20.
In the first part of the thesis we propose a modified version of the transfer matrix method, based on the parallel version of the Montecarlo algorithm, allowing us to go to NR = 100.
In the second part we present EnaS, a C++ library with a Graphical User Interface developed for neuroscientists. EnaS offers highly interactive tools that allow users to manage data, perform empirical statistics, modeling and visualizing results.
Finally, in a third part, we test our method on synthetic and real data sets. Real data set correspond to retina data provided by neuroscientists partners. Our non extensive analysis shows the advantages of considering spatio-temporal correlations for the analysis of retina spike trains, but it also outlines the limits of Maximum Entropy methods.
For more information about the software that I co-developed with my colleagues, please visit this page:
https://enas.inria.fr/
For more information about the publications, please visit this page:
https://scholar.google.fr/citations?user=L97ZODwAAAAJ
For the thesis, please visit this link:
https://www.theses.fr/178166669
Statically Balanced Tensegrity Mechanisms By SchenkTensegrity Wiki
The document summarizes a graduation colloquium on the theory and design of statically balanced tensegrity mechanisms. The objectives of the graduation project were to develop a fundamental understanding of zero stiffness tensegrity structures and generate design guidelines through theoretical research and building a demonstration model. Key findings included that zero stiffness requires the directions of conventional members to lie on a conic, and the number of zero stiffness modes is equal to 6 minus the number of unique bar directions on the conic. A prototype tensegrity structure was designed and built to demonstrate the properties of a statically balanced mechanism.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
- The document recaps fundamentals of finite element analysis (FEA) theory, including loads, deformation, stress, strain, Hooke's law, Poisson ratio, and 3D Hooke's law.
- It also reviews matrix notation, operations like addition/subtraction, multiplication by a scalar, transposition, matrix multiplication, and properties of the inverse of a matrix.
This document provides an introduction and examples of calculus, Taylor's theorem, and matrix. It discusses the use of calculus, matrices, and Taylor's theorem in engineering fields like civil, electrical, and computer science. Examples of integration, differentiation, and matrix operations are included. The document also references textbooks for further information.
A Displacement-Potential Scheme to Stress Analysis of a Cracked Stiffened Pan...inventionjournals
This paper deals with an efficient analytical scheme for the analysis of stress and displacement fields of boundary-value problemsof plane elasticity with mixed boundary conditions and material discontinuity. More specifically, the mechanical behavior of a stiffened panel with an edge crack is analyzed under the influence of axial loading, using a new analytical scheme. Earlier mathematical models of elasticity were very deficient in handling the practical stress problems of solid mechanics. Analytical methods of solution have not gained that much popularity in the field of stress analysis, mainly because of the inability of dealing with mixed boundary conditions, irregular boundary shapes, and material discontinuity
Paper Study: Melding the data decision pipelineChenYiHuang5
Melding the data decision pipeline: Decision-Focused Learning for Combinatorial Optimization from AAAI2019.
Derive the math equation from myself and match the same result as two mentioned CMU papers [Donti et. al. 2017, Amos et. al. 2017] while applying the same derivation procedure.
Similar to Composite Systems - Trace Approach _ PPT Presentation (20)
This document provides a summary of Filipe Amorim Gonçalves Giesteira's resume. It lists his personal details such as age, citizenship, licenses and contact information. It then outlines his work experience in mechanical engineering roles focusing on adhesive bonding and joining technologies. It also details his education including a Master's degree in mechanical engineering from the University of Porto with a specialization in structural engineering and machine design. His areas of interest and skills in languages, software and hobbies are also listed.
A thorough review of the theoretical background of the invariant based approach to composite designs. The trace material property was explained, along with several topics from linear elasticity theory. The MATLAB implementation of a tool capable of determining the laminate stiffness properties from the lamina properties and stacking sequence was also presented. For access to the tool please contact the author: filipegiesteira@outlook.com
Assignment developed in the scope of the Composite Systems course, lecture at FEUP (Faculdade de Engenharia da Universidade do Porto).
The theoretical formulation of an isoparametric element, from the Lagrange family. In addition, the MATLAB code of the FEM from the bilinear element with four nodes was also implemented.
Assignment developed in the scope of the finite element method course, lectured at FEUP (Faculdade de Engenharia da Universidade do Porto).
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The theoretical formulation of an isoparametric element, from the Lagrange family. In addition, the MATLAB code of the FEM from the bilinear element with four nodes was also implemented.
Assignment developed in the scope of the finite element method course, lectured at FEUP (Faculdade de Engenharia da Universidade do Porto).
Report regarding the design of a mechanical speed reducer, using two steps with cylindrical gears and one step with bevel gears. Assignment conducted during the Introduction to Machine Design course, lectured at FEUP (Faculty of Engineering of University of Porto)
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Equation summary with the main formulae and procedures used to design screw elements. Both fixing screws and transmission screws were considered.
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Equation summary with the main formulae and procedures used to design to Fatigue.
Lecture notes developed during the Machine Elements I course, lectured at FEUP (Faculdade de Engenharia da Universidade do Porto).
Este documento discute conceitos fundamentais de dinâmica de sistemas mecânicos. Aborda tópicos como geometria de massas, equações diferenciais, componentes energéticas, teoremas dinâmicos e análise de movimentos básicos. Também apresenta sistemas de um e dois graus de liberdade, sistemas contínuos como cordas, barras e vigas, além de métodos aproximados para resolução de problemas de vibração.
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Specification drawing of a shaft with multiple sections, used to assemble two helical gears and two ball bearings. The assembly of the gears was done using the mechanism hub keyway-keyway-shaft keyway
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2. 2
• Theory of Linear Elasticity – Basic overview and
review of the most important aspects
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
3. Theory of Elasticity
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
3
• Cauchy Stress Tensor
𝜎 =
𝜎𝑥𝑥 𝜏 𝑥𝑦 𝜏 𝑥𝑧
𝜏 𝑦𝑥 𝜎 𝑦𝑦 𝜏 𝑦𝑧
𝜏 𝑧𝑥 𝜏 𝑧𝑦 𝜎𝑧𝑧
Valid according to the following assumptions
– Continuum Medium (macroscopic analysys)
– Homogeneous Material - specific properties are
indepedent of the point of evalution, i.e.
Mechancial properties of any given point are equal do the
specific properties of the solid
𝜏 𝑦𝑥
𝜏 𝑥𝑦
𝑦
𝑥𝜏 𝑦𝑥
𝜏 𝑥𝑦
11. Theory of Elasticity
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
11
• Coordinate system
Transformation
𝑇 =
𝑙 𝑥′ 𝑙 𝑦′ 𝑙 𝑧′
𝑚 𝑥′ 𝑚 𝑦′ 𝑚 𝑧′
𝑛 𝑥′ 𝑛 𝑦′ 𝑛 𝑧′
=
cos 𝛼 𝑥′ 𝑥 cos 𝛼 𝑥′ 𝑦 cos 𝛼 𝑥′ 𝑧
cos 𝛼 𝑦′ 𝑥 cos 𝛼 𝑦′ 𝑦 cos 𝛼 𝑦′ 𝑧
cos 𝛼 𝑧′ 𝑥 cos 𝛼 𝑧′ 𝑦 cos 𝛼 𝑧′ 𝑧
[ 𝑇 ]
= −
cos 𝜓 sin 𝜓 0
sin 𝜓 cos 𝜓 0
0 0 1 𝑧
1 0 0
0 cos 𝜃 sin 𝜃
0 − sin 𝜃 cos 𝜃 𝑥
−
cos 𝜑 sin 𝜑 0
sin 𝜑 cos 𝜑 0
0 0 1 𝑧
Euler Angles
12. Theory of Elasticity
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
12
• Coordinate system
Transformation
𝜎 = 𝑇 𝜎 𝑇 𝑇
𝜀 = 𝑇 𝜀 𝑇 𝑇
𝑧
𝑥
𝑦
𝑥′
𝛼 𝑥′ 𝑥
𝛼 𝑥′ 𝑦
𝛼 𝑥′ 𝑧
14. Theory of Elasticity
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
14
• Orthotropic Materials
Material Symmetry
𝑒3
𝑒2
𝑒1
𝑒3
𝑒2
𝑒1
𝑒3
𝑒2
𝑒1
18. Classical Laminated Plate
Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
18
• Laminate
– Series of superimposed plies, fabrics or mats, building a thin layered
body
– Stacking process (draping or lay-up)
• Ply Concept
– Semi-finished product with reinforcement(fiber)
and matrix
– Can be understood as a thin quasi 2D geometry
19. Classical Laminated Plate
Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
19
• Laminate – Building Blocks
– Ply Fabric Mat
20. Classical Laminated Plate
Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
20
• Laminate
– Ply Fabric Mat
21. Classical Laminated Plate
Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
21
• Model of UD Ply
2D Orthotropic Governing Equations
𝜎 = 𝑄 𝜀
𝜎1
𝜎2
𝜏6
=
𝑄11 𝑄12 𝑄16
𝑄21 𝑄22 𝑄26
𝑄61 𝑄62 𝑄66
𝜀1
𝜀2
𝜀6
𝜎1
𝜎2
𝜏6
=
𝐸1
1 − 𝑣12 𝑣21
𝐸1 𝑣12
1 − 𝑣12 𝑣21
0
𝐸2 𝑣21
1 − 𝑣12 𝑣21
𝐸2
1 − 𝑣12 𝑣21
0
0 0 𝐺66
𝜀1
𝜀2
𝜀6
4 Independent
elastic coefficients
Symmetry
Relations:
𝒗 𝟏𝟐
𝒗 𝟐𝟏
=
𝑬 𝟐𝟐
𝑬 𝟏𝟏
Engineering Notation – Based on the angular distortion
22. Classical Laminated Plate
Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
22
• Model of UD Ply
2D Orthotropic Stress and Strains tensors
σ =
𝜎11 𝜏12
𝜏21 𝜎22
; ε =
𝜀1 𝜀12
𝜀21 𝜀22
• Coordinate system Transformation
𝑇 =
cos 𝛼 𝑥′ 𝑥 cos 𝛼 𝑥′ 𝑦 cos 𝛼 𝑥′ 𝑧
cos 𝛼 𝑦′ 𝑥 cos 𝛼 𝑦′ 𝑦 cos 𝛼 𝑦′ 𝑧
cos 𝛼 𝑧′ 𝑥 cos 𝛼 𝑧′ 𝑦 cos 𝛼 𝑧′ 𝑧
[ 𝑡 ] =
cos 𝜓 sin 𝜓
−sin 𝜓 cos 𝜓 𝑧
In ply plane Rotation
23. Classical Laminated Plate
Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
23
• Coordinate system
Transformation
𝜎 = 𝑡 𝜎 𝑡 𝑇
Basic Algebraic Manipulation
𝜎 = 𝑡σ 𝜎
𝜎1′
𝜎2′
𝜏6′
=
cos2 𝜃 sin2 𝜃 −2 sin 𝜃 cos 𝜃
sin2
𝜃 cos2
𝜃 2 sin 𝜃 cos 𝜃
sin 𝜃 cos 𝜃 −sin 𝜃 cos 𝜃 cos2 𝜃 − sin2 𝜃
𝜎1
𝜎2
𝜏6
24. Classical Laminated Plate
Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
24
• Coordinate system
Transformation
Basic Algebraic Manipulation
𝜀 = 𝑡 𝜀 𝑡 𝑇
ε = 𝑡 𝜀 ε
𝜀1′
𝜀2′
𝛾6′
=
cos2 𝜃 sin2 𝜃 − sin 𝜃 cos 𝜃
sin2 𝜃 cos2 𝜃 sin 𝜃 cos 𝜃
2 sin 𝜃 cos 𝜃 −2 sin 𝜃 cos 𝜃 cos2 𝜃 − sin2 𝜃
𝜀1
𝜀2
𝛾6
26. Invariant-based Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
26
• Trace = 1st Invariant or
Linear Invariant
𝑇𝑟 = 𝑄11 + 𝑄22 + 2𝑄66
Trace of the on-axis ply plane
Stiffness Matrix
Trace is a material property
Trace is Independent from the
axis (by definition of 1st
invariant)
𝑥1
𝑧3
𝑦2
27. Invariant-based Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
27
• Trace = 1st Invariant or
Linear Invariant
𝑇𝑟 = 𝑄11 + 𝑄22 + 2𝑄66
Only the stress-strain relation in
terms of tensorial stress and
strain has the invariant tensor
properties
𝑥1
𝑧3
𝑦2
33. Invariant-based Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
33
• Master Ply Concept
Laminate testing - Properties as-built,
allowing to take in account for:
• Lay-up technique and associate flaws
and typical human variation
• Interface
• Particular Matrix Material
• Curing conditions
𝑬 𝟏𝟏
∗
≈ 0.8796
• Fiber dominated
Stiffness Coefficient
• Less Matrix/process
dependent
• More accuracy in the
Trace evaluation
34. Invariant-based Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
34
• Application of Master Ply
Concept
Ply
Characterization
Coupon Testing (UD Ply)
Laminate Testing
Laminate Design
35. Invariant-based Theory
• Theory of Elasticity (Review)
• Classical Laminated Plate Theory
• Invariant-based Theory
• MATLAB Tool
35
• Application of Master Ply
Concept – Coupon Testing
𝑬 𝟏𝟏
∗
=
𝑬 𝟏𝟏
𝑻𝒓
⇒ 𝑻𝒓 =
𝑬 𝟏𝟏
𝑬 𝟏𝟏
∗ =
𝑬 𝟏𝟏
𝟎.𝟖𝟕𝟗𝟔
𝑬 𝟏𝟏 empirical evaluation
46. References
46
• Tsai, S.W. and J.D.D. Melo, An invariant-based theory of composites.
Composites Science and Technology, 2014. 100: p. 237-243.
• Nettles, A.T., Basic mechanics of laminated composite plates. 1994.
• Roylance, D., Laminated composite plates. Massachusetts Institute of
Technology Cambridge, 2000.
• Tavakoldavani, K., Composite materials equivalent properties in lamina,
laminate, and structure levels. 2014: The University of Texas at Arlington.
• Liu, S. and W. Su, Effective couple-stress continuum model of cellular solids
and size effects analysis. International Journal of Solids and Structures, 2009.
46(14): p. 2787-2799.
• Irgens, F., Continuum Mechanics. 2008: Springer Berlin Heidelberg.
• Seth, B., Generalized strain measure with applications to physical problems.
1961, WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER.
47. References
47
• Slaughter, W.S., The Linearized Theory of Elasticity. 2002: Birkhäuser Boston.
• Peck, S. Invariant-Based Design of Laminated Composite Materials. in 50th
AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials
Conference 17th AIAA/ASME/AHS Adaptive Structures Conference 11th AIAA
No.
• Theory of elasticity. 2001: McGraw-Hill.
• Moakher, M., The algebra of fourth-order tensors with application to
diffusion MRI. Visualization and Processing of Tensor Fields, 2009: p. 57-80.
• Sih, G.C., A. Carpinteri, and G. Surace, Advanced Technology for Design and
Fabrication of Composite Materials and Structures: Applications to the
Automotive, Marine, Aerospace and Construction Industry. 1995: Springer
Netherlands.
• Gay, D., Composite Materials: Design and Applications, Third Edition. 2014:
Taylor & Francis.