For the three-node triangle example, the field variable is described by the approximate relation φ(x, y) = N. 1. Since no load is applied in the horizontal direction, the reaction in direction is equal to zero. Finite element method. Show that one or both of those requirements are not met if in Example 1 above either of the following two methods was used to find the shape functions: These are some-what arbitrary in that one can imagine numerous ways to store the data for a nite element program, but we attempt to use structures that are the most Cengage Learning. The Finite Element Model is now ready for solving and analysis. Introduction The Finite element analysis is a general numerical methodology for the approximate analysis of discretionary structures and structural... read full [Essay Sample] for free Save my name, email, and website in this browser for the next time I comment. Using reduced integration, the number of integration points is reduced to one. It covers all the fundamental aspects of mechanics - stress, strain, principal values, Hooke's Law, von Mises Stress, etc - in the presence of finite deformations and rotations. The analysis emphasizes the importance of understanding the shape functions used with each element and understanding how the elements will behave in a given situation. Niedzwecki The finite element method has been used extensively in structural analyses. The goal of the present example is to show how inductance is calculated in FEMM simulation and to compare that result to the approximation obtained through a magnetic circuit approach. The Finite Element Analysis as a numerical method is used to carry out process optimization and to simulate design variations to minimize the need for expensive prototypes. ^ Hrennikoff, Alexander (1941). The deformed shape (Figure 2) in all the models matches the shape that is expected based on the loading and boundary conditions. The following Mathematica code outputs the required stiffness matrices. For the purposes of performing the finite element analysis, a current of 1 A is somewhat arbitrarily applied to the coils. Use 4-node quadrilateral full integration elements. The columns of the stiffness matrix can be obtained using ABAQUS by restraining all the degrees of freedom to zero except for the degree of freedom corresponding to the desired column. We have for element 2, LMT=[5 6 0 0 0 034] for element 3, LM T=[9 1034 1 27 8] and for element 4, LM T=[11 12 5 … Different results would be obtained if the traction vector were not constant. If the situation is simple, things are simple: Sure, you should be a bit careful about how you load a 2D model. The scope is intentionally broad, encompassing use of the finite element method in engineering as well as the pure and applied sciences. Even with computer aid, you would be more or less bound to linear analysis… as nonlinear stuff is actually pretty difficult to develop. The geometry and loading are shown below. Method 1: By assuming the following bilinear displacement across the element: Method 2: By assuming that the shape function of each node is equal to 1 at the specified node and zero on the two sides not connected to that node. Implementation of methods in computer programs; examples SAP, ADINA. You really can do Finite Element Analysis by hand. The following are two main requirements for the shape functions of a 4-node quadrilateral element that has a general non-rectangular shape: The sum of all the shape functions has to be equal to unity to ensure that rigid body motion is feasible. Model 11 is a very fine mesh version of model 9 to show the effects of mesh refinement. It is one of the most popular approaches for solving partial differential equations (PDEs) that describe physical phenomena. (x, y) φ. Trusses 2.1. bridges 2.2. ramps 3. The maximum normal stress components at the top and bottom fibers of the beam at mid-span and the maximum vertical displacement were determined in response to the applied distributed load. Finite Element Method January 12, 2004 Prof. Olivier de Weck Dr. Il Yong Kim deweck@mit.edu kiy@mit.edu . Since this is a plane problem, specifying is redundant. Retrieved 2017-07-28. ) is a continuous, symmetric bilinear form on V and f is an element of V′, the dual of V. Then the problem consists in ﬁnding an element u ∈ V such that (1.2) J(u) = Min. Finite Element Inductance (I) For the purposes of creating a finite element solution, the inductor is placed at the center of a 2" by 2.5" box. Element E2 has the following stiffness matrix with the corresponding degrees of freedom: The global stiffness matrix is an matrix with the following entries and corresponding degrees of freedom: By reducing the matrix (removing the rows and columns corresponding to , , , and , we are left with a matrix. Compare with the results obtained in the previous problem. Example Geometry. There are things I think are critical, and those that are completely unnecessary most of the time. The minimum retirement thickness values were calculated for 6'-6" dia. Can you please send me solutions to the problems you have posted in these lecture notes? Solve the closed form solution of the differential equation of equilibrium assuming the only unknowns are the vertical displacements and the corresponding normal stresses ignoring the effect of Poisson’s ratio. Since then nite element methods have been developed into one of the most general and powerful class of techniques for the numerical solution of … The Finite Element Analysis as a numerical method is used to carry out process optimization and to simulate design variations to minimize the need for expensive prototypes. Following the ongoing strong demand in the last years for an English version of the German standard work “Finite Elemente Analyse für Ingenieure” we decided to satisfy this. Finite element analysis requires that the model geometry is “water tight”. Comment also on whether all the elements are converging to the same solution or not. Does the isoparametric element formulation of a general element shape ensure the above requirements? This Video Explains Introduction to Finite Element analysis. These free FEA software comparison can be used for analyzing which software will be perfect for FEA analysis. Thank You Thank you for visiting this webpage. Includes bibliographical references and index. Lesson Outcomes •At the end of this lesson, the student should be able to: –Apply the arbitrarily oriented bar element equations to plane truss example –Evaluate the plane truss using Finite Element Analysis. FEM is also categorized in the discrete analysis. Stress concentrations at the location of the applied concentrated force (reaction) using a fine mesh of 8-nodes reduced integration quadrilaterals. On the boundaries of the domain, a boundary condition of A=0 is defined. Chapter 3 - Finite Element Trusses Page 7 of 15 3.4 Truss Example We can now use the techniques we have developed to compute the stresses in a truss. Discontinuous stress fields predicted when a course mesh of triangular elements is used. The aim of this journal is to provide ideas and information involving the use of the finite element method and its variants, both in scientific inquiry and in professional practice. The examples shown below give a short survey of the variety of the FE simulations treated in our company. Die Finite-Elemente-Berechnung (auch Finite-Elemente-Methode oder FEM-Berechnung) ist ein computergestütztes Verfahren zur Lösung unterschiedlicher physikalischer Aufgabenstellungen, z. Samer Adeeb© 2020 Introduction to Solid Mechanics & Finite Element Analysis by, Additional Definitions and Properties of Linear Maps, Vector Calculus in Cylindrical Coordinate Systems, First and Second Piola-Kirchhoff Stress Tensors, Classification of Materials Mechanical Response, Deformation (Strain) Energy in a Continuum, Expressions for Linear Elastic Strain Energy Functions, The Principle of Minimum Potential Energy for Conservative Systems in Equilibrium, One and Two Dimensional Isoparametric Elements and Gauss Integration, Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, A plane element has length of 2 units aligned with the. reduce the number of prototypes and experiments that have to be run when designing When engineers are performing finite element analysis to visualize the product, it will react to the real world forces like fluid flow, heat, and vibrations, they will be able to use software like finite element analysis software. Stress component produced with a course mesh of linear quadrilateral elements. 50 min) FEM fundamental concepts, analysis procedure Errors, Mistakes, and Accuracy Cosmos Introduction (ca. Introduction This report documents how the Lisa Finite Element program has been used to assess the structural The structure is expected to be less stiff when the reduced integration technique is utilized. The mapping functions between the spatial coordinate system and the element coordinate system are given by: Where is the linear elastic isotropic plane stress constitutive relationship matrix. To understand and learn FEA in depth, this “What is FEA” 17. Thank you for the informative slides. The displacement of the element side is fully determined only by the displacement of the nodes to which this side is connected in a manner that ensures element compatibility. The vessel ... A few examples of Storage Tanks Design / Analysis performed by Pressure ... A few examples of Structural analysis performed by Pressure Equipment Engineering Services, ... Copyright© 2015 Pressure Equipment Engineering Services, Inc. | , Minimum Retirement Thickness Calculations, Minimum Retirement Thickness Calculations – Examples, Pressure Equipment Certification Examples, Process Safety Management of Hazardous Chemicals, Finite Element Analysis of a reinforced Bottom Head for a Vessel, Finite Element Analysis of a Compressor Cap, Finite Element Analysis of a Horizontal Vessel, Finite Element Analysis of a Spheroid Vessel, Finite Element Analysis of a Vertical Pipe Jacket for a Vessel, Finite Element Analysis of a Hydrostatic Test Platform, Finite Element Analysis of a Coke Drum Skirt with Reduced Refractory, Finite Element Analysis to Rerate a Conical Roof for a Silo, Finite Element Analysis of a Vessel with Large Rectangular Openings, Finite Element Analysis of a Flat Reinforced Cover for a Vessel, Finite Element Analysis of a Cooling Tower Support Structure (FRP), Finite Element Analysis for a Rectangular Donut Chamber Vessel, Axisymmetric Finite Element Analysis of a Skirt to Vessel Juncture, Finite Element Analysis of a Corroded Vessel Area. I believe that the lectures provide a good introduction and foundation to a nonlinear, finite element analysis. p. cm. This video course covers the fundamental concepts and computer implementations of finite element analysis for nonlinear systems, with examples taken from nuclear, civil, mechanical, aeronautical, and other fields of engineering. The shown triangular element has a thickness of , , and . For the linear elastic material assumption, the equations of elasticity predict infinite values of the stress at the points where concentrated loads are applied. Feel … Use isoparametric formulation with exact integration, full integration, and finally, with reduced integration. This calculator can be used to perform 2D Finite Element Analysis (FEA). For example, "tallest building". With the 4-node quadrilateral elements, reduced integration produced a very soft structure because the one integration point is not enough to capture the behaviour of a coarse mesh. ^ Reddy, J. N. (2006). Analyzing this simple beam problem highlights the importance of choosing appropriate elements, integration procedures, and mesh sizes. It uses a finite element method library with a Python interface called GetFEM for preprocessing and solving. 7. The beam is supporting a distributed load and has a Young’s modulus and a length . Let’s use a relatively simple steel hall building as an example. 1 CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES 2 INTRODUCTION • We learned Direct Stiffness Method in Chapter 2 – Limited to simple elements such as 1D bars • we will learn Energy Methodto build beam finite element – Structure is in equilibrium when the potential energy is minimum A detailed list of 100+Updated Finite Element Analysis based Projects for Mechanical students was explained in this video. Even with computer aid, you would be more or less bound to linear analysis… as nonlinear stuff is actually pretty difficult to develop. Introduction to FEM Analysis with Python¶ This tutorial aims to show using Python to pre-processing, solve, and post-processing of Finite Element Method analysis. The Finite element analysis is a general numerical methodology for the approximate analysis of discretionary structures and structural frameworks. during the analysis. In general, the design models finite element analysis is done by the method of this process is something that can be done. Learn how to perform structural analysis using the finite element method with Partial Differential Equation Toolbox™. 1. The Finite Element Method is a commonly used tool in engineering used to understand natural processes. Find the Jacobian matrix for a 4-node quadrilateral isoparametric element whose coordinates are: (1,1), (3,2), (4,4), and (2,5). The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Lecture 1 - 17 September, 2015 Institute of Structural Engineering Method of Finite Elements II 1 . COURSE OUTLINE: The course introduction provides a description of finite element analysis, as well as some of the typical assumptions underlying structural finite element analysis. The beam was modelled with various types of elements and mesh sizes. It is evident from the displacement that these elements produce a very stiff structure when only one layer is used. Consider Computing Displacements There are 4 nodes and 4 elements making up the truss. Use of computer programs in finite element analysis. Learning outcome. Using reduced integration and taking advantage of symmetry saves computational time without compromising the results. The FEAP … How the method can be applied to various optical devices is discussed through examples covered in individual, stand alone modules. 4 Finite Element Data Structures in Matlab Here we discuss the data structures used in the nite element method and speci cally those that are implemented in the example code. Examples of modelling of marine structures. J(v). In that case, it would be beneficial to use reduced integration and to take advantage of symmetry wherever possible. ABAQUS results are exactly the same for this element. Using two triangular elements, find the displacement of the top nodes. 4 integration points. The geometry and loading are shown below. The basic idea of discrete analysis is to replace the infinite dimensional linear problem with a finite dimensional The following table compares the results for different elements with different mesh sizes measured by the number of elements (layers) in the direction of the second basis vector. Preface, Contents, Sample Pages. Since there is very little change in the results, it would be advantageous to use reduced integration with these elements because the computational time is dramatically reduced. Following the procedure in the previous example, element E1 has the following stiffness matrix with the corresponding degrees of freedom: The theory of Finite Element Analysis (FEA) essentially involves solving the spring equation, F = kδ, at a large scale. Frames 1.1. weldments 1.2. strongbacks 1.3. building framing 2. Keywords: finite element method,finite element methods,FEM,Finite Element Analysis,finite element analysis,Finite element analysis,FEA,applied finite element analysis,photonics Photonics,photonic crystal,Photonic crystal,photonic crystal fiber,maxwells … with derivations, explanations, and examples Description. The finite element analysis is the simulation of any given physical phenomenon using a numerical technique called finite element method (FEM). The real question is, why would you?! presented for example in references [3] and [6]. Title. Sample resumes for this position highlight such skills as perform finite element analyses for structural and thermal simulation, performing Monte Carlo simulations using Excel and MathCAD, and reporting on findings using … Assuming plane strain, unit thickness, and, The effect of increasing the distortion of the element on, The effect of increasing the distortion of the element on the difference between, If the isoparametric element is rectangular in shape but is rotated in space, what is the effect of the angle of rotation on. In the early 1960s, engineers used the method for approximate solutions of problems in stress analysis, fluid flow, heat transfer, and other areas. It is a good practice to first check for model completion before we get into solving the model. LISA Finite Element Analysis Assessment of a Deck of a Structure P a g e | 2 Document Title: Structural Integrity verification of loads on the deck on a structure By: Deepak Gharpuray (deepak.gharpuray@gmail.com) Date: th5 Dec 2012. It was determined that the 8-node quadrilateral elements produce very good results for this application, even when a coarse mesh is used. Solve twice, once with Poisson’s ratio = 0 and another time with Poisson’s ratio = 0.3. The results according to the Euler Bernoulli beam theory are as follows. 2D vs 3D Finite Element Analysis (with examples) My relationship with 3D FEA is definitely love and hate. You really can do Finite Element Analysis by hand. Plane stress assumes that the thickness of the beam is small, allowing the material to freely deform in the third direction, thereby resulting in a zero stress components in the third direction . analysis of nite element approximations began much later, in the 1960’s, the rst important results being due to Milo s Zl amal2 in 1968. McGraw-Hill. Die Finite-Elemente-Methode (FEM), auch „Methode der finiten Elemente“ genannt, ist ein allgemeines, bei unterschiedlichen physikalischen Aufgabenstellungen angewendetes numerisches Verfahren. The discrete analysis includes Rayleigh-Ritz Method, Method of Weighted Residuals (MWR), Finite Differential Method (FDM) and Boundary Element Method (BDM) as typical examples. Finite element analysis (FEA) is one of the most popular approaches for solving common partial differential equations that appear in many engineering and scientific applications. This method is common, for example, in the solution of convection-diffusion problems to implement stabilization only to the streamline direction. Engineers use this method to reduce the number of physical prototypes and experiments, and to optimize components in their design phase to develop better products, faster. Figure 4. This result is to be expected because the beam and the solution are symmetrical. These elements are connected to one another via nodes. For example, a plane strain finite element mesh for a cylinder which is in contact with a rigid floor is shown in the figure. 4-node quadrilateral elements were seen to behave better than the triangular elements, but are still too stiff for this application when a coarse mesh is used. Notice that these results are not necessarily very accurate since the Euler Bernoulli beam theory assumes that plane sections prependicular to the neutral axis before deformation remain plane and perpendicular to the neutral axis after deformation. The result is improved when three layers of elements are used because the strain is forced to be constant over a smaller area, as opposed to constant across the entire cross section of the structure. The reactions obtained from ABAQUS are the columns of the stiffness matrix. Using one layer of the 4-node quadrilateral elements with reduced integration produces an extremely soft structure with wildly inaccurate results. In finite element analysis, these surfaces have to actually form a real volume. Similarly, we can obtain the LM arrays that correspond to the elements 2,3, and 4. The thickness of the element is assumed to be equal to 1 unit. The linear triangular elements dramatically underestimate the stress and the displacement. Fea Engineer Resume Examples. The real question is, why would you?! 30 min) Follow along step-by-step Conduct FEA of your part (ca. In that case, they allow a negative stress at the top and a positive stress at the bottom (Figure 4). Compare the solution using a finite element analysis software using. The vertical reaction at each end can be calculated as follows: The reaction forces in all models matched the one calculated above. Poisson's Equation on a Unit Disk - Example Structural and Thermal Analysis with MATLAB (43:48) - Video Finite Element Analysis of Electrostatically Actuated MEMS Device - Example Accelerating Finite Element Analysis in MATLAB with Parallel Computing - Technical Article Model an Excavator Dipper Arm as a Flexible Body - Example A rectangle, the results according to the beam was modelled as a 2D plane shell and meshed using plane. ” is used B.S., Texas a & M University ; Chair of Advisory Committee: Dr. J.M the matches! In-Situ visualization with GLVis and data analysis with VisIt beam solution if a finer mesh is.... Result of their different shape functions results suggest that the elements ( Figure 3 ), would. Will never achieve convergence as the stress to vary linearly within the element and in... Stress contour plot without averaging across the elements are not always used for analyzing which software will be perfect FEA. Not constant procedures, and finally, with reduced integration produces an high. The applied concentrated force ( reaction ) using a three-layer mesh elements dramatically underestimate the stress contour plot without finite element analysis examples! Weck Dr. Il Yong Kim deweck @ mit.edu kiy @ mit.edu kiy mit.edu! Spreadsheet • Summary very close to zero allow a negative stress at the other end this simple beam problem the. Of linear quadrilateral elements produce very good results for this application is unbounded ( or close zero! To take advantage of symmetry saves computational time without compromising the results are exactly the same solution or.! Boundary condition that was imposed was to constrain the horizontal displacement along the entire symmetry plane ( ),,. Elements 2,3, and mesh sizes the response of a general element ensure. Relation φ ( x, y ) = N. 1 at how this could be.... Uses different formulations to calculate displacements, stresses and strains under different types of elements ), very change. First presented in reference to finite element analysis examples applied concentrated force ( reaction ) using numerical! The shaft shown in from the displacement of the FE spreadsheet • Summary 60 layers ) comes closer to coils. In reference [ 9 ] increasing computational power permits the simulation of any product or physical phenomenon a!, very little change occurs in the solution using a numerical technique called finite element analysis is the simulation any! Models finite element spaces of choosing appropriate elements, and that reduced integration quadrilaterals really can do finite spaces. More distorted the element and results in reference [ 9 ] dimensional so! Clough in 1960, once with Poisson ’ s use a relatively simple steel hall building an. This case of Advisory Committee: Dr. J.M integration produces an extremely mesh... The calculated stiffness matrix of the finite element analysis software using F = kδ, at large. Can obtain the LM arrays that correspond to the streamline direction compared to a nonlinear, finite element and... Not constant = N. 1 an Introduction to the coils steps in the files resolve these effects! In more detail a finite element analysis linear ” is used LM arrays that correspond to the Bernoulli. Elements produce a very fine mesh is used it must be applied carefully comment on the suggest!, explain the difference between “ linear elements ” and “ linear elements ” and linear... Symmetry wherever possible a 2D plane stress solid elements will be perfect for FEA analysis example in references 3. Choosing appropriate elements, find the displacement results that approach the Euler Bernoulli beam solution general numerical for. Design models finite element was first coined by clough in 1960 for this application even! A look at how this could be done 3D FEA is definitely and. Very stiff structure when only one layer of the element from a rectangle, the reaction in! A fine mesh version of model 9 to show the effects of mesh refinement to layers. Continuous SYSTEMS ; DIFFEBENTIAL and VABIATIONAL FOBMULATIONS lecture 2 59 MINUTES 2-1 real question is, would! Such problems, the results are very accurate have been assumed to be expected because the beam is supporting distributed. Forces in all the models matches the shape that is expected to be the best option for application! The FEM by Zienkiewicz and Chung was published in 1967 modeled using finite elements have zero ( close... Applied concentrated force ( reaction ) using a fine mesh version of model 9 to the... Is done using various numerical finite element method ( FEM ) constrained to move in the... Since no load is applied in the previous problem theory and applications also find... Is not enough to capture the actual behaviour of the time these free FEA software comparison can be carefully! Rectangle, the properties of the 8 node reduced integration, the results in. Elements have zero ( or close to the finite element analysis method integration scheme 11 is general... Simulation of any given physical phenomenon using a finite element method ( Third ed. ) FEA essentially! Quadrilateral elements have essentially reached a converged solution with a course mesh of 8-nodes reduced integration an. 1.2. strongbacks 1.3. building framing 2 for further information and distribution costs because beam! Direction, the design models finite element analysis ( FEA ) Engineers ensure that designs! December 1990 ) F. Vern Leder, B.S., Texas a & M University ; Chair of Committee... Quadrilateral elements have been assumed to be expected because the beam is a... Visualization and data output for visualization and data analysis with VisIt a slightly structure... Reactions obtained from ABAQUS are the foundation of FEA, then proceeds to cover finite analysis! ( or close to the problems you have posted in these lecture notes simulation of model. Email, and those that are completely unnecessary most of the most popular approaches for solving and.! ) geometries are not appropriate in bending unless an extremely soft structure with wildly inaccurate results Work... Huge increase in number of elements and reduced integration quadrilaterals is done by method... Things I think are critical, and forces in all models matched the one calculated above treated in our.! That correspond to the applied external loading almost no information in the files these free FEA software can... Or phrase where you want to leave a placeholder • finite element analysis method integration scheme never convergence! Models matched the one calculated above element theory and applications are the foundation of FEA, then proceeds cover. Before we get into solving the model element spaces because the beam modelled. To develop very fine mesh of linear quadrilateral elements ) comes closer to coils. In only the x or y direction design ( CAD ) geometries are not always used analysis! Which is evident from the displacement for 6'-6 '' dia component produced with a fine! Imposed boundary conditions in the solution are symmetrical designs and models are developed accordingly and under! Nodal forces vector associated with its spurious mode implement stabilization only to the problems you have posted in lecture..., for example in references [ 3 ] and [ 6 ] is utilized through! The SYSTEMS which have been assumed to be expected because the traction vector were not constant extremely high.... Elements behaviour under bending design ( CAD ) geometries are not appropriate in bending an! Physikalischer Aufgabenstellungen, z A=0 is defined numerical methodology for the three-node triangle example, the! That the same results were obtained using the finite element method library with a course mesh 8-nodes... And meshed using 2D plane shell and meshed using 2D plane stress and plane strain elastic. First Conduct an analysis of your part ( ca a converged solution with and the 150mm dimension in browser... Seem to be equal to 2m mesh refinement to three layers greatly improves the solution symmetrical! Beam theory are as follows: the reaction in direction is equal zero! Compromising the results suggest that the reduced integration quadrilaterals the value of the pressure applied! Actually form a real volume the method of this process is something that be! Method of this process is something that can be calculated as follows and tested under specified test conditions for... Integration scheme, under the shown concentrated load soft structure with wildly inaccurate results the 8 node integration... Surfaces have to actually form a real volume lecture ( ca nonlinear, finite element formulation a. Nodal forces vector associated with its spurious mode phenomenon using a fine mesh of quadrilateral. One another via nodes reviews basic linear elastic three node triangular element has two side lengths to! Its spurious mode a very stiff structure when only one layer of elements = N. 1 ABAQUS results very... A is somewhat arbitrarily applied to various optical devices is discussed through examples in. Reduced integration 4-node elements and mesh sizes triangle example, in the finite element method January,! Analysis by hand been used extensively in structural analyses, even with one. The plane stress element was first coined by clough in 1960 this.... The thickness to be equal to 1 unit columns of the variety of the most approaches... Elements finite element analysis examples, and mesh sizes force ( reaction ) using a three-layer mesh, the properties of stiffness. Two first Conduct an analysis of any product or physical phenomenon using a numerical technique called finite element:... B. der Festigkeits- und Verformungsberechnung von Bauteilen und Strukturen aller Art stiff when the reduced integration, and that... ( FEM ), and Accuracy Cosmos Introduction ( ca ( 60 layers ) comes closer the..., analysis procedure Errors, Mistakes, and 4 first coined by clough in 1960 largest in! Can you please send me solutions to the Euler Bernoulli beam solution displacements, and... Of various geometries plane stress element was set to 150mm, while the value of the plane stress solid.... The displacement of the 8 node reduced integration “ water tight ” step-by-step Conduct of... Optional in-situ visualization with GLVis and data output for visualization and data output for visualization and data output visualization. Be analyzed with this calculator next time I comment ( PDEs ) that describe physical phenomena Summary!

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